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<b>let </b><font color="Maroon">L1</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L2</font> be    <a href="orders_2.html#L1">RelStr</a> , <font color="Maroon">L3</font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">f1</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font>;<br/>



<b>assume </b><a NAME="E1:6"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f1</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>let </b><font color="Maroon">f2</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L3</font>;<br/>

<b>assume </b><a NAME="E2:6"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f2</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>let </b><font color="Maroon">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font>;<br/>

<b>assume </b><a NAME="E3:6"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:6_1"/>
( ( not <font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a> ) or <font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a> or <font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_1"><b>suppose </b></a><a NAME="E1:6_1_1"/>
( not <font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a> &amp; not <font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">L1</font> = <font color="Maroon">L1</font>, <font color="Maroon">L2</font> = <font color="Maroon">L2</font>, <font color="Maroon">L3</font> = <font color="Maroon">L3</font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/><b>reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font> ;<br/><b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f2</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L3</font> ;<br/><b>consider </b><font color="Maroon">g1</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L1</font><b> such that </b><br/><a NAME="E6:6_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">g1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K2">"</a> 
 <b>and </b><br/><a NAME="E7:6_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">g1</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/><a NAME="E8:6_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E9:6_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f1</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">g2</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L3</font>,<font color="Maroon">L2</font><b> such that </b><br/><a NAME="E11:6_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K2">"</a> 
 <b>and </b><br/><a NAME="E12:6_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">g2</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/><a NAME="E13:6_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E14:6_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">f2</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/><a NAME="E15:6_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T46">FUNCT_1:46</a></i>;<br/><a NAME="E16:6_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T14">YELLOW_2:14</a></i>;<br/><a NAME="E17:6_1_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g2</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i>, , <a class="ref" href="yellow_2.html#T14">YELLOW_2:14</a></i>;<br/><a NAME="E18:6_1_1"/>
<font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="funct_1.html#K2">"</a> 
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#T66">FUNCT_1:66</a></i>;<br/><b>hence </b><a NAME="E19:6_1_1"/>
<font color="Maroon">h</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_2"><b>suppose </b></a><a NAME="E1:6_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:6_1_2"/><b>then </b>
<font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:6_1_2"/><b>then </b>
<font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E4:6_1_2"/>
<font color="Maroon">h</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_3"><b>suppose </b></a><a NAME="E1:6_1_3"/>
<font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:6_1_3"/><b>then </b>
( <font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i>, , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E3:6_1_3"/>
<font color="Maroon">h</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/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="6_1_4"><b>suppose </b></a><a NAME="E1:6_1_4"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:6_1_4"/><b>then </b>
<font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:6_1_4"/><b>then </b>
<font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E4:6_1_4"/>
<font color="Maroon">h</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/></div><b>end;</b></div></div><b>end;</b></div>

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