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<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font>, <font color="Maroon" title="c3">L3</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>given </b><font color="Maroon" title="c4">f1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font><b> such that </b><a NAME="E1:16"/><i><font color="Green" title="E12">A1</font></i>: 
<font color="Maroon" title="c4">f1</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a><br/>

<b>given </b><font color="Maroon" title="c5">f2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">L2</font>,<font color="Maroon" title="c3">L3</font><b> such that </b><a NAME="E2:16"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c5">f2</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">WAYBEL_1:def 8</a><br/>

<div><i><font color="Green" title="E14">A3</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" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L1</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/><b>hence </b><a NAME="E3:16_1"/>
<font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f1</font> is   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T19" title="FUNCT_2:th.19">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" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> ) or <font color="Maroon" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> or <font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">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" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; not <font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c6">L1</font> = <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c7">L2</font> = <font color="Maroon" title="c2">L2</font>, <font color="Maroon" title="c8">L3</font> = <font color="Maroon" title="c3">L3</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  ;<br/><b>reconsider </b><font color="Maroon" title="c9">f1</font> = <font color="Maroon" title="c4">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c6">L1</font>,<font color="Maroon" title="c7">L2</font> ;<br/><b>reconsider </b><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c5">f2</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">L2</font>,<font color="Maroon" title="c8">L3</font> ;<br/><b>consider </b><font color="Maroon" title="c11">g1</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">L2</font>,<font color="Maroon" title="c6">L1</font><b> such that </b><br/><a NAME="E6:16_2_1"/><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c11">g1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>and </b><br/><a NAME="E7:16_2_1"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c11">g1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E8:16_2_1"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c9">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E9:16_2_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c9">f1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">g2</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c8">L3</font>,<font color="Maroon" title="c7">L2</font><b> such that </b><br/><a NAME="E11:16_2_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>and </b><br/><a NAME="E12:16_2_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c12">g2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E13:16_2_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c10">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E14:16_2_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c10">f2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E15:16_2_1"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E8:16_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="waybel_1.html#E13:16_2_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="funct_1.html#T46" title="FUNCT_1:th.46">FUNCT_1:46</a></i>;<br/><a NAME="E16:16_2_1"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E9:16_2_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="waybel_1.html#E14:16_2_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="yellow_2.html#T14" title="YELLOW_2:th.14">YELLOW_2:14</a></i>;<br/><a NAME="E17:16_2_1"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c11">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">g2</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E7:16_2_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="waybel_1.html#E12:16_2_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="yellow_2.html#T14" title="YELLOW_2:th.14">YELLOW_2:14</a></i>;<br/><a NAME="E18:16_2_1"/>
<font color="Maroon" title="c11">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c12">g2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font></span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 <b>by </b><i><a class="txt" href="waybel_1.html#E6:16_2_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="waybel_1.html#E8:16_2_1"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="waybel_1.html#E11:16_2_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="waybel_1.html#E13:16_2_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="funct_1.html#T66" title="FUNCT_1:th.66">FUNCT_1:66</a></i>;<br/><a NAME="E19:16_2_1"/><b>then </b>
<font color="Maroon" title="c10">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E15:16_2_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="waybel_1.html#E16:16_2_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="waybel_1.html#E17:16_2_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E20:16_2_1"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">Def8</a></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" title="E15">A15</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_2"/><b>then </b>
<font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:16_2_2"/><b>then </b><i><font color="Green" title="E16">A16</font></i>: 
<font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <b>by </b><i><a class="txt" href="waybel_1.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="waybel_1.html#E1:16_2_2"><i><font color="Green" title="E15">A15</font></i></a></i>;<br/><a NAME="E5:16_2_2"/>
<font color="Maroon" title="c6">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16_2_2"><i><font color="Green" title="E15">A15</font></i></a>, <a class="txt" href="waybel_1.html#E3:16_2_2"><i><font color="Green" title="E16">A16</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E6:16_2_2"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">Def8</a></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" title="E15">A17</font></i>: 
<font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_3"/><b>then </b><i><font color="Green" title="E16">A18</font></i>: 
( <font color="Maroon" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> &amp; <font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> )
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel_1.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="waybel_1.html#E1:16_2_3"><i><font color="Green" title="E15">A17</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E4:16_2_3"/>
<font color="Maroon" title="c6">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16_2_3"><i><font color="Green" title="E16">A18</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E5:16_2_3"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">Def8</a></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" title="E15">A19</font></i>: 
<font color="Maroon" title="c3">L3</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 ;<br/><div class="add"><a NAME="E2:16_2_4"/><b>then </b><i><font color="Green" title="E16">A20</font></i>: 
<font color="Maroon" title="c2">L2</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E2:16"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:16_2_4"/><b>then </b><i><font color="Green" title="E17">A21</font></i>: 
<font color="Maroon" title="c1">L1</font> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <b>by </b><i><a class="txt" href="waybel_1.html#E1:16"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="waybel_1.html#E3:16"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="waybel_1.html#E2:16_2_4"><i><font color="Green" title="E16">A20</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E5:16_2_4"/>
<font color="Maroon" title="c6">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel_1.html#E1:16_2_4"><i><font color="Green" title="E15">A19</font></i></a>, <a class="txt" href="waybel_1.html#E3:16_2_4"><i><font color="Green" title="E17">A21</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/><b>hence </b><a NAME="E6:16_2_4"/>
<font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c3">L3</font> <a href="waybel_1.html#R5" title="WAYBEL_1:pred.5">are_isomorphic</a> 
 <b>by </b><i><a class="ref" href="waybel_1.html#D8" target="_self" title="WAYBEL_1:def.8">Def8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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