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

<span class="kw">let </span><font color="Maroon" title="c1">F</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Z</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">r</font> being    <a href="field_5.html#M1" title="FIELD_5:mode.1">Renaming</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>,<font color="Olive" title="b1">Z</font> holds <br/> ( <font color="Olive" title="b2">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Olive" title="b2">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b2">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Olive" title="b2">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b2">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Z</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">r</font> being    <a href="field_5.html#M1" title="FIELD_5:mode.1">Renaming</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">Z</font> holds <br/> ( <font color="Olive" title="b1">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Olive" title="b1">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b1">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Olive" title="b1">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">r</font> be    <a href="field_5.html#M1" title="FIELD_5:mode.1">Renaming</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">F</font>,<font color="Maroon" title="c2">Z</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">K</font> =  <a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font>;<br/>
<a NAME="E1:68"/><span class="lab"><font color="Green" title="E27">H0</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">r</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="c1">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D5" target="_self" onmouseover="rs('field_5/D5')" onmouseout="rh()">defr</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">b</font></span>)</span> <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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">b</font></span>)</span> <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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">r</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D8" target="_self" onmouseover="rs('field_5/D8')" onmouseout="rh()">defrf</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K4" title="FIELD_5:func.4">ren_add</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D8" target="_self" onmouseover="rs('field_5/D8')" onmouseout="rh()">defrf</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D6" target="_self" onmouseover="rs('field_5/D6')" onmouseout="rh()">defra</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E27">H0</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:68"/>
<font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="vectsp_1.html#V13" title="VECTSP_1:attr.13">additive</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="68_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font></span>)</span> <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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> <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="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c3">r</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D8" target="_self" onmouseover="rs('field_5/D8')" onmouseout="rh()">defrf</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="field_5.html#K5" title="FIELD_5:func.5">ren_mult</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D8" target="_self" onmouseover="rs('field_5/D8')" onmouseout="rh()">defrf</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D7" target="_self" onmouseover="rs('field_5/D7')" onmouseout="rh()">defrm</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E27">H0</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E5:68"/>
<font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<a NAME="E6:68"/>
<font color="Maroon" title="c3">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <span class="p1">(<span class="default"><a href="field_5.html#K6" title="FIELD_5:func.6">RenField</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#D8" target="_self" onmouseover="rs('field_5/D8')" onmouseout="rh()">defrf</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:68"/>
<font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="group_1.html#V6" title="GROUP_1:attr.6">unity-preserving</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E27">H0</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E8:68"/>
<font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/>

<span class="kw">thus </span><a NAME="E9:68"/>
<font color="Maroon" title="c3">r</font> <a href="field_5.html#K7" title="FIELD_5:func.7">"</a>  is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="field_5.html#T27" target="_self" onmouseover="rs('field_5/T27')" onmouseout="rh()">lemonto</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
