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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">ADD</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E25">A1</font></i>: 
for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being  <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">f1</font>,<font color="Olive" title="b2">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Olive" title="b2">f2</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">F</font> =  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<b>thus </b><a NAME="E2:45"/>
<font color="Maroon" title="c2">ADD</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<a NAME="E1:45_1"/>
 ex <font color="Olive" title="b1">map'</font> being   <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  st <font color="Olive" title="b1">map'</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">ADD</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">map0</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">map0'</font> = <font color="Maroon" title="c4">map0</font> as    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">map0'</font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">f</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><a NAME="E1:45_1_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">map0'</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E1:45_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">map0'</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">map0'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></i>;<br/><a NAME="E2:45_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">map0'</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">map0'</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:45_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">map0'</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:45_1_1_1_1"/>
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">map0'</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:45_1_1_1"/>
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">map0'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E1:45_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">map0'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">map0'</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></i>;<br/><a NAME="E2:45_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">map0'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:45_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">map0'</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:45_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">map0'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:45_1_1_1_2"/>
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">map0'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:45_1_1_1"/>
( <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c4">map0</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c4">map0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:45_1_1_1"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45_1_1"/>
<font color="Maroon" title="c5">map0'</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c2">ADD</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_1"/>
<font color="Maroon" title="c2">ADD</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:45"/>
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a><br/>

<a NAME="E1:45_2"/>
( <font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c4">f1</font> &amp; <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:45_2"/>
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c4">f1</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:45"/>
<font color="Maroon" title="c2">ADD</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">f3</font> be    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a><br/>

<b>reconsider </b><font color="Maroon" title="c7">ADD12</font> = <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c8">ADD23</font> = <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">f3</font> as   <a href="pscomp_1.html#NM1" title="PSCOMP_1:NM.1">RealMap</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<a NAME="E2:45_3"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/><a NAME="E1:45_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> )
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></i>;<br/><a NAME="E2:45_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span>
 ;<br/><a NAME="E3:45_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f3</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></i>;<br/><b>hence </b><a NAME="E4:45_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:45_3_1"/>
<font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:45_3"/>
( <font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">f2</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c8">ADD23</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">ADD12</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>
<a NAME="E4:45_3"/><b>then </b>
( <font color="Maroon" title="c4">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c8">ADD23</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">ADD12</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font> &amp; <font color="Maroon" title="c7">ADD12</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c6">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c7">ADD12</font>,<font color="Maroon" title="c6">f3</font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="nagata_1.html#E2:45_3"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:45_3"/>
<font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c5">f2</font>,<font color="Maroon" title="c6">f3</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">f1</font>,<font color="Maroon" title="c5">f2</font></span>)</span>,<font color="Maroon" title="c6">f3</font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:45"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div>
