<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">R1</font>, <font color="Maroon" title="c2">R2</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>   <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">N1</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b3">N2</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b2">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">N2</font> holds <br/>for <font color="Olive" title="b4">J1</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">N1</font>,<font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b5">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b4">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b4">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b5">J2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N1</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b2">N2</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b1">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">N2</font> holds <br/>for <font color="Olive" title="b3">J1</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N1</font>,<font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b4">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b3">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b3">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b4">J2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">N1</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">R1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N2</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font>  st <font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">N2</font> holds <br/>for <font color="Olive" title="b2">J1</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b3">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b2">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b2">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b3">J2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">N2</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c2">R2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N2</font> implies for <font color="Olive" title="b1">J1</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b2">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b1">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b1">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b2">J2</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:88"/><span class="lab"><font color="Green" title="E58">A1</font></span>: 
<font color="Maroon" title="c4">N1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">N2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">J1</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c1">R1</font><br/> for <font color="Olive" title="b2">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Olive" title="b1">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b1">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b2">J2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">J1</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,<font color="Maroon" title="c1">R1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">J2</font> being   <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c2">R2</font>  st <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">J2</font> holds <br/> <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Olive" title="b1">J2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">J2</font> be    <a href="yellow_6.html#M3" title="YELLOW_6:mode.3">net_set</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">N2</font>,<font color="Maroon" title="c2">R2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J2</font> implies  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:88"/><span class="lab"><font color="Green" title="E59">A2</font></span>: 
<font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">J2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span><br/>

<a NAME="E3:88"/><span class="lab"><font color="Green" title="E60">A3</font></span>: 
 <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 <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c4">N1</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></span>;<br/>
<a NAME="E4:88"/><span class="lab"><font color="Green" title="E61">A4</font></span>: 
 <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 <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c5">N2</font>,<span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c7">J2</font></span>)</span></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">g</font> = the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E62">A5</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E61">A4</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
;<br/>
<a NAME="E6:88"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> holds <br/>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> implies the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:88_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span><br/>

<a NAME="E2:88_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span>
 
;<br/>
<a NAME="E3:88_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">N1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">x1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">N1</font>, <font color="Maroon" title="c12">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c6">J1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:88_2"/><span class="lab"><font color="Green" title="E63">A6</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="yellow_3.html#K7" title="YELLOW_3:func.7">[</a><span class="default"><font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font></span><a href="yellow_3.html#K7" title="YELLOW_3:func.7">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">y1</font> = <font color="Maroon" title="c11">x1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">y2</font> = <font color="Maroon" title="c12">x2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K5" title="YELLOW_1:func.5">product</a> <font color="Maroon" title="c7">J2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a></span>;<br/>
<span class="kw">thus </span>the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c11">x1</font>,<font color="Maroon" title="c12">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_2"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c6">J1</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c11">x1</font></span>)</span> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">x2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c11">x1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></span>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c7">J2</font> <a href="yellow_6.html#K7" title="YELLOW_6:func.7">.</a> <font color="Maroon" title="c13">y1</font></span>)</span> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">y2</font> <a href="waybel_3.html#K4" title="WAYBEL_3:func.4">.</a> <font color="Maroon" title="c13">y1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a></span>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">y1</font>,<font color="Maroon" title="c14">y2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D16" title="YELLOW_6:def.16">YELLOW_6:def 16</a></span>
<br/>.= 
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88_2"><span class="lab"><font color="Green" title="E63">A6</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E7:88"/><span class="kw">then </span>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <span class="p1">(<span class="default"><a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:88"><span class="lab"><font color="Green" title="E62">A5</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:88"/>
 <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c6">J1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_6.html#K8" title="YELLOW_6:func.8">Iterated</a> <font color="Maroon" title="c7">J2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E58">A1</font></span></a>, <a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E59">A2</font></span></a>, <a class="txt" href="#E3:88"><span class="lab"><font color="Green" title="E60">A3</font></span></a>, <a class="txt" href="#E4:88"><span class="lab"><font color="Green" title="E61">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
