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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#V2">reflexive</a> <a href="orders_2.html#V3">transitive</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#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/><b>hence </b><a NAME="E2:19_1"/>
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>consider </b><font color="Maroon">g</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><br/><a NAME="E4:19_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Maroon">g</font> is <a href="orders_3.html#V5">monotone</a> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as  <a href="orders_3.html#V5">monotone</a> <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> = <font color="Maroon">g</font>;<br/><a NAME="E6:19_1"/>
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><b>hence </b><a NAME="E7:19_1"/>
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="funct_2.html#T35">FUNCT_2:35</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:19"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 ;<br/>

<b>given </b><font color="Maroon">g</font> being  <a href="orders_3.html#V5">monotone</a> <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="E3:19"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T</font> &amp; <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font> )
 ;<br/>

<a NAME="E4:19"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T29">FUNCT_2:29</a></i>;<br/>
<a NAME="E5:19"/><b>then </b>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i>, <a class="ref" href="funct_2.html#T36">FUNCT_2:36</a></i>;<br/>
<b>hence </b><a NAME="E6:19"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/>


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