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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S2</font> #) &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font> #) implies for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font>  st <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> holds <br/>for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b2">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )</span><br/>

<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E4">A1</font></i>: 
(  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">S1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">S2</font> #) &amp;  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T1</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T1</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">T2</font> #) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font>  st <font color="Olive" title="b1">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> holds <br/>for <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b2">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> holds <br/><font color="Olive" title="b2">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a></span><br/>

<b>let </b><font color="Maroon" title="c5">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">T1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> implies for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )</span><br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green" title="E5">A2</font></i>: 
<font color="Maroon" title="c5">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> holds <br/><font color="Olive" title="b1">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a></span><br/>

<b>let </b><font color="Maroon" title="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c4">T2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font> implies <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )</span><br/>

<b>assume </b><a NAME="E3:5"/><i><font color="Green" title="E6">A3</font></i>: 
<font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a></span><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" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> or  not <font color="Maroon" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a></span><br/><div class="add"><a NAME="E2:5_1_1"/><b>then </b>
( <font color="Maroon" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c3">T1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 <b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E5">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="E3:5_1_1"/><b>then </b>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">T2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/><a NAME="E4:5_1_1"/><b>then </b>
( <font color="Maroon" title="c2">S2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> &amp; <font color="Maroon" title="c4">T2</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 ;<br/><b>hence </b><a NAME="E5:5_1_1"/>
<font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c1">S1</font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c7">S1</font> = <font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c8">T1</font> = <font color="Maroon" title="c3">T1</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E5">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="E3:5_1_2"/>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c7">S1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">S2</font> = <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c10">T2</font> = <font color="Maroon" title="c4">T2</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a>  <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c5">f</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c7">S1</font>,<font color="Maroon" title="c8">T1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c6">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c9">S2</font>,<font color="Maroon" title="c10">T2</font> ;<br/><a NAME="E7:5_1_2"/><i><font color="Green" title="E7">A4</font></i>: 
( <font color="Maroon" title="c11">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">T1</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Olive" title="b2">y</font> ) ) )
 <b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">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" title="c13">x</font>, <font color="Maroon" title="c14">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">S2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font> iff <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">y</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c15">a</font> = <font color="Maroon" title="c13">x</font>, <font color="Maroon" title="c16">b</font> = <font color="Maroon" title="c14">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c7">S1</font> <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E4">A1</font></i></a></i>;<br/><a NAME="E2:5_1_2_1"/>
( ( <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font> implies <font color="Maroon" title="c15">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">b</font> ) &amp; ( <font color="Maroon" title="c15">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c16">b</font> implies <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font> ) &amp; ( <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">y</font> implies <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c16">b</font> ) &amp; ( <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c15">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c11">f</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c16">b</font> implies <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">y</font> ) )
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E3:5"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="yellow_0.html#T1" title="YELLOW_0:th.1">YELLOW_0:1</a></i>;<br/><b>hence </b><a NAME="E3:5_1_2_1"/>
( <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">y</font> iff <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c13">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c12">g</font> <a href="yellow_6.html#K3" title="YELLOW_6:func.3">.</a> <font color="Maroon" title="c14">y</font> )
 <b>by </b><i><a class="txt" href="#E2:5"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E9:5_1_2"/>
<font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="#E1:5"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="#E3:5"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="#E7:5_1_2"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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