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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>given </b><font color="Maroon">f1</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font><b> such that </b><a NAME="E1:16"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">f1</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" target="_self">WAYBEL_1:def 8</a><br/>

<b>given </b><font color="Maroon">f2</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L2</font>,<font color="Maroon">L3</font><b> such that </b><a NAME="E2:16"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f2</font> is <a href="waybel_0.html#V23">isomorphic</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" target="_self">WAYBEL_1:def 8</a><br/>

<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L1</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> is   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T19">FUNCT_2:19</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_2"/>
( ( 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="16_2_1"><b>suppose </b></a><a NAME="E1:16_2_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="E6:16_2_1"/><i><font color="Green">E48</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="E7:16_2_1"/><i><font color="Green">E49</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="E8:16_2_1"/><i><font color="Green">E50</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="E9:16_2_1"/><i><font color="Green">E55</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="E11:16_2_1"/><i><font color="Green">E57</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="E12:16_2_1"/><i><font color="Green">E58</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="E13:16_2_1"/><i><font color="Green">E59</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="E14:16_2_1"/><i><font color="Green">E60</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="E15:16_2_1"/><i><font color="Green">E61</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="E16:16_2_1"/><i><font color="Green">E62</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="E17:16_2_1"/><i><font color="Green">E63</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="E18:16_2_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/><a NAME="E19:16_2_1"/><b>then </b>
<font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</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/><b>hence </b><a NAME="E20:16_2_1"/>
<font color="Maroon">L1</font>,<font color="Maroon">L3</font> <a href="waybel_1.html#R5" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_2"><b>suppose </b></a><a NAME="E1:16_2_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">L1</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_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="E3:16_2_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<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>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font> <b>by </b><i>, </i>;<br/><a NAME="E5:16_2_2"/>
<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/><b>hence </b><a NAME="E6:16_2_2"/>
<font color="Maroon">L1</font>,<font color="Maroon">L3</font> <a href="waybel_1.html#R5" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_3"><b>suppose </b></a><a NAME="E1:16_2_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">L2</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_3"/><b>then </b><i><font color="Green">E67</font></i>: 
( <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>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font> <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E4:16_2_3"/>
<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/><b>hence </b><a NAME="E5:16_2_3"/>
<font color="Maroon">L1</font>,<font color="Maroon">L3</font> <a href="waybel_1.html#R5" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_2_4"><b>suppose </b></a><a NAME="E1:16_2_4"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">L3</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_4"/><b>then </b><i><font color="Green">E69</font></i>: 
<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="E3:16_2_4"/><b>then </b><i><font color="Green">E70</font></i>: 
<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>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L3</font> <b>by </b><i>, , , <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E5:16_2_4"/>
<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/><b>hence </b><a NAME="E6:16_2_4"/>
<font color="Maroon">L1</font>,<font color="Maroon">L3</font> <a href="waybel_1.html#R5" target="_self">are_isomorphic</a> 
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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