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

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ADD</font> being   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)</span>)</span>  st (  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">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="Olive" title="b1">ADD</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Olive" title="b2">f1</font>,<font color="Olive" title="b3">f2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f1</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Olive" title="b3">f2</font> ) holds <br/>( <font color="Olive" title="b1">ADD</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Olive" title="b1">ADD</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Olive" title="b1">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">F</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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/>
<span class="kw">let </span><font color="Maroon" title="c3">ADD</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <span class="p1">(<span class="default"><a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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="c3">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> ) implies ( <font color="Maroon" title="c3">ADD</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:45"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 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="c3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">ADD</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a>  &amp; <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )</span><br/>

<a NAME="E2:45"/>
 ex <font color="Olive" title="b1">map9</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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">map9</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">ADD</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T45" title="FUNCOP_1:th.45">FUNCOP_1:45</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">map09</font> = <font color="Maroon" title="c4">map0</font> as    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>) <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">map09</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">map09</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">ADD</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">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="c3">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> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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">map09</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></span><br/><a NAME="E1:45_1_1_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">map09</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">map09</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></span>;<br/><a NAME="E2:45_1_1_1"/><span class="kw">then </span>
<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">map09</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1_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">map09</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E2:45_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">f</font> <a href="nagata_1.html#K3" title="NAGATA_1:func.3">+</a> <font color="Maroon" title="c5">map09</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/><span class="kw">hence </span><a NAME="E3:45_1_1"/>
( <font color="Maroon" title="c3">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="c3">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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:45_1"/>
<font color="Maroon" title="c5">map09</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c3">ADD</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="binop_1.html#T3" title="BINOP_1:th.3">BINOP_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:45"/>
<font color="Maroon" title="c3">ADD</font> is  <a href="setwiseo.html#V1" title="SETWISEO:attr.1">having_a_unity</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setwiseo.html#D2" title="SETWISEO:def.2">SETWISEO:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>  &amp; <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E4:45"/>
<font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D13" title="BINOP_1:def.13">BINOP_1:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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="c3">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>)</span><br/>

<a NAME="E1:45_2"/>
<font color="Maroon" title="c3">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:45_2"/>
<font color="Maroon" title="c3">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="c3">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:45"/>
<font color="Maroon" title="c3">ADD</font> is  <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">f1</font>, <font color="Maroon" title="c5">f2</font>, <font color="Maroon" title="c6">f3</font> be    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>); <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="binop_1.html#D14" title="BINOP_1:def.14">BINOP_1:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">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="c3">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="c3">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">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>)</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">ADD12</font> = <font color="Maroon" title="c3">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="c3">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/>
<a NAME="E2:45_3"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<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="c3">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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></span><br/><a NAME="E1:45_3_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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></span>;<br/><a NAME="E2:45_3_1"/><span class="kw">then </span>
<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"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></span>;<br/><span class="kw">hence </span><a NAME="E4:45_3_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nagata_1.html#D7" target="_self" title="NAGATA_1:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:45_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>;<br/>
<a NAME="E5:45_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a></span>;<br/>
<a NAME="E6:45_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E4:45_3"><span class="lab"><font color="Green" title="E27">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:45_3"/>
<font color="Maroon" title="c3">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="c3">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="c3">ADD</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c3">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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:45_3"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
