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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="orders_2.html#L1">RelStr</a> ;<br/>

<b>assume </b><a NAME="E1:5"/><i><font color="Green">E5</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></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">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></font> #) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/>

<b>assume </b><a NAME="E3:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> or  not <font color="Maroon">c<sub>1</sub></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="5_1_1"><b>suppose </b></a><a NAME="E1:5_1_1"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><a NAME="E2:5_1_1"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="struct_0.html#V3">empty</a> )
 <b>by </b><i><a class="txt" href="waybel29.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:5_1_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="waybel29.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><a NAME="E4:5_1_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is <a href="struct_0.html#V3">empty</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="struct_0.html#V3">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="E5:5_1_1"/>
<font color="Maroon">c<sub>6</sub></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="5_1_2"><b>suppose </b></a><a NAME="E1:5_1_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>3</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="waybel29.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="waybel_0.html#D38">WAYBEL_0:def 38</a></i>;<br/><a NAME="E3:5_1_2"/>
( the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a>  <b>by </b><i><a class="txt" href="waybel29.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> ;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> ;<br/><a NAME="E7:5_1_2"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>8</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> iff <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="waybel29.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>9</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="txt" href="waybel29.html#E1:5"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E2:5_1_2_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> ) &amp; ( <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> implies <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> ) &amp; ( <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> ) &amp; ( <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>16</sub></font> implies <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="waybel29.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="waybel29.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="yellow_0.html#T1">YELLOW_0:1</a></i>;<br/><b>hence </b><a NAME="E3:5_1_2_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> iff <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="yellow_6.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="waybel29.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E9:5_1_2"/>
<font color="Maroon">c<sub>6</sub></font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel29.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="waybel29.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="waybel29.html#E7:5_1_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="waybel_0.html#T66">WAYBEL_0:66</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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