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

<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">M</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> <br/>  for <font color="Olive" title="b2">f</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">M</font>  ex <font color="Olive" title="b3">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Olive" title="b1">M</font> st <br/>( <font color="Olive" title="b3">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b3">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">M</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</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">f</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">M</font>  ex <font color="Olive" title="b2">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b2">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">M</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">  ex <font color="Olive" title="b1">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b1">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Olive" title="b1">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) );<br/>
<a NAME="E1:87"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">y</font> =  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>);<br/>
<span class="kw">take </span>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</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">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)]</span><br/>

<span class="kw">thus </span><a NAME="E3:87_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>, <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c5">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c4">F1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E3:87"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  &amp; (  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E46">A1</font></span></a>);<br/></span>
<a NAME="E4:87"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" onmouseover="rs('funct_2/T8')" onmouseout="rh()">FUNCT_2:8</a></span>;<br/>
<a NAME="E5:87"/>
<font color="Maroon">S<sub>1</sub></font>[1,<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/>
<a NAME="E6:87"/><span class="kw">then </span>
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E7:87"/><span class="kw">then </span>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<font color="Maroon" title="c1">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E8:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A4</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E9:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A5</font></span>: 
<font color="Maroon" title="c3">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c4">F1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">X1</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c4">F1</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">fs</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fs</font> holds <br/>( ( (  for <font color="Olive" title="b2">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> holds <br/><font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) implies ( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> holds <br/> ex <font color="Olive" title="b3">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b4">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b5">m1</font>, <font color="Olive" title="b6">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b7">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b8">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b6">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m1</font> &amp; <font color="Olive" title="b8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">m2</font> &amp; <font color="Olive" title="b3">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b7">f1</font>,<font color="Olive" title="b8">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b3">fs1</font> ) ) ) ) &amp; (  ex <font color="Olive" title="b2">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> &amp; <font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) implies $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) );<br/>
<a NAME="E11:87"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  holds <br/> ex <font color="Olive" title="b2">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  implies  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">fs</font> = <font color="Maroon" title="c6">e</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_2_1"/>
(  for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> holds <br/><font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> or  ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> &amp; <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_2_1_1"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> holds <br/><font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><a NAME="E2:87_2_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E3:87_2_1_1"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E4:87_2_1_1"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:87_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A8</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_2_1_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_2_1_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_2_1_1_1_1"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">u</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ;<br/><span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:87_2_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1_1_1"><span class="lab"><font color="Green" title="E54">A9</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_2_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A10</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">u</font> = <font color="Maroon" title="c3">f</font>;<br/><span class="kw">take </span>
<font color="Maroon" title="c3">f</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">f</font>]</span><br/><span class="kw">thus </span><a NAME="E2:87_2_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">f</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1_1_2"><span class="lab"><font color="Green" title="E54">A10</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:87_2_1_1_1_3"/><span class="lab"><font color="Green" title="E54">A11</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">n</font> =  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">n9</font> = <font color="Maroon" title="c8">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><a NAME="E4:87_2_1_1_1_3"/>
<font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A11</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><a NAME="E5:87_2_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
<font color="Maroon" title="c9">n9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/><a NAME="E6:87_2_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A13</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">n9</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T3" onmouseover="rs('afinsq_1/T3')" onmouseout="rh()">AFINSQ_1:3</a></span>;<br/><span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> );<br/><a NAME="E7:87_2_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A14</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font> holds <br/> ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font> implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Olive" title="b1">x</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2_1_1_1_3_1"/><span class="lab"><font color="Green" title="E58">A15</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</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">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Olive" title="b1">x</font>]</span><br/>

<a NAME="E2:87_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A16</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<a NAME="E3:87_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E4:87_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/>
<a NAME="E5:87_2_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A18</font></span>: 
<font color="Maroon" title="c8">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">k</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">m2</font> = <font color="Maroon" title="c8">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c10">k</font> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="txt" href="#E5:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A18</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m1</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c11">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E1:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E6:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a></span>;<br/>
<a NAME="E9:87_2_1_1_1_3_1"/>
 <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="xreal_1.html#T24" onmouseover="rs('xreal_1/T24')" onmouseout="rh()">XREAL_1:24</a></span>;<br/>
<a NAME="E10:87_2_1_1_1_3_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E11:87_2_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">m2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E4:87_2_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">f2</font> = <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">m2</font> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c12">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E6:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">x</font> = <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>;<br/>
<span class="kw">take </span>
<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</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">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font>,<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>]</span><br/>

<span class="kw">thus </span><a NAME="E13:87_2_1_1_1_3_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">k</font>,<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c13">f1</font>,<font color="Maroon" title="c14">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</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">consider </span><font color="Maroon" title="c10">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E9:87_2_1_1_1_3"/><span class="lab"><font color="Green" title="E58">A19</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font> &amp; (  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c9">n9</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c10">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="finseq_1.html#S1" onmouseover="rs('finseq_1/S1')" onmouseout="rh()">FINSEQ_1:sch 1</a>(<a class="txt" href="#E7:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E57">A14</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c11">u</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font>;<br/><span class="kw">take </span>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font>]</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_1_1_1_3_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">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> holds <br/><font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) implies ( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> ) ) ) )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:87_2_1_1_1_3_2"/>
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> holds <br/><font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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"> ( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> ) ) )</span><br/><span class="kw">thus </span><a NAME="E2:87_2_1_1_1_3_2"/>
( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E54">A11</font></span></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">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> )</span><br/><span class="kw">thus </span><a NAME="E3:87_2_1_1_1_3_2"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_2_1_1_1_3_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">n99</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c12">n99</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> implies  ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_2_1_1_1_3_2_1"/>
<font color="Maroon" title="c12">n99</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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"> (  not  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> or  ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:87_2_1_1_1_3_2_1"/><span class="lab"><font color="Green" title="E59">A20</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</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">  ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c10">fs1</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"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; <font color="Maroon" title="c10">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E3:87_2_1_1_1_3_2_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E2:87_2_1_1_1_3_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="txt" href="#E9:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E58">A19</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</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"> ( (  for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; <font color="Maroon" title="c10">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp;  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> )</span><br/>

<span class="kw">thus </span><a NAME="E4:87_2_1_1_1_3_2_1"/>
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n99</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; <font color="Maroon" title="c10">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_2_1_1_1_3_2_1"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a>, <a class="txt" href="#E5:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="txt" href="#E9:87_2_1_1_1_3"><span class="lab"><font color="Green" title="E58">A19</font></span></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">  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font></span><br/>

<span class="kw">thus </span><a NAME="E5:87_2_1_1_1_3_2_1"/>
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</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"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:87_2_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>, <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c10">fs1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_1"><span class="lab"><font color="Green" title="E52">A7</font></span></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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_2_1_2"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
 ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c7">fs</font> &amp; <font color="Maroon" title="c7">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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">  ex <font color="Olive" title="b1">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Olive" title="b1">u</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c3">f</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">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">f</font>]</span><br/><span class="kw">thus </span><a NAME="E2:87_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">e</font>,<font color="Maroon" title="c3">f</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_2_1_2"><span class="lab"><font color="Green" title="E52">A21</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">F2</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E13:87"/><span class="lab"><font color="Green" title="E52">A22</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  &amp; (  for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="classes1.html#S1" onmouseover="rs('classes1/S1')" onmouseout="rh()">CLASSES1:sch 1</a>(<a class="txt" href="#E11:87"><span class="lab"><font color="Green" title="E51">A6</font></span></a>);<br/></span>
<a NAME="E14:87"/><span class="lab"><font color="Green" title="E53">A23</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">fs</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; (  for <font color="Olive" title="b3">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b2">fs</font> holds <br/><font color="Olive" title="b2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) &amp;  ex <font color="Olive" title="b3">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b4">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b4">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b5">m1</font>, <font color="Olive" title="b6">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b7">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b8">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b6">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b5">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b6">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">p</font> &amp; <font color="Olive" title="b7">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m1</font> &amp; <font color="Olive" title="b8">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">m2</font> &amp; <font color="Olive" title="b3">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b7">f1</font>,<font color="Olive" title="b8">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b3">fs1</font> ) holds <br/><font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">fs</font> being   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font>  st <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> &amp; (  for <font color="Olive" title="b2">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Olive" title="b1">fs</font> holds <br/><font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) &amp;  ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> ) holds <br/><font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c8">fs</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</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="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> &amp; (  for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> holds <br/><font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) &amp;  ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) implies <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:87_3"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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"> (  not  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> or  ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> &amp; <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) or  for <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  ex <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><br/>  for <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>  holds <br/>(  not <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> or  not <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> or  not <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) ) or  not <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) or <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E2:87_3"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</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"> (  ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> &amp; <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) or  for <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  ex <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><br/>  for <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>  holds <br/>(  not <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> or  not <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> or  not <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) ) or  not <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) or <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E3:87_3"/>
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> holds <br/><font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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"> (  for <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds <br/>(  not  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 or  ex <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><br/>  for <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>  holds <br/>(  not <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> or  not <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> or  not <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> or  not <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) ) or  not <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> ) or <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) )</span><br/>

<span class="kw">assume </span><a NAME="E4:87_3"/>
 ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E6:87_3"/><span class="lab"><font color="Green" title="E55">A25</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E7:87_3"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">and </span><br/><a NAME="E8:87_3"/><span class="lab"><font color="Green" title="E57">A27</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c9">fs1</font>
 ;<br/>
<a NAME="E9:87_3"/><span class="lab"><font color="Green" title="E58">A28</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> holds <br/> ex <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3_1"><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">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> implies  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</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">  ex <font color="Olive" title="b1">y</font>, <font color="Olive" title="b2">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Olive" title="b2">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<a NAME="E2:87_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:87_3_1"/><span class="lab"><font color="Green" title="E59">A29</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Y</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:87_3_1"/><span class="lab"><font color="Green" title="E60">A30</font></span>: 
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c11">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3_1"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">p</font> = <font color="Maroon" title="c12">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a></span>;<br/>
<a NAME="E8:87_3_1"/>
<font color="Maroon" title="c13">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:87_3_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c13">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">m1</font>, <font color="Maroon" title="c15">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c16">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c14">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c17">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E11:87_3_1"/><span class="lab"><font color="Green" title="E61">A31</font></span>: 
( <font color="Maroon" title="c14">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c15">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> &amp; <font color="Maroon" title="c17">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c17">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">y</font>, <font color="Maroon" title="c19">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E13:87_3_1"/><span class="lab"><font color="Green" title="E62">A32</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c19">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3_1"><span class="lab"><font color="Green" title="E59">A29</font></span></a>, <a class="txt" href="#E6:87_3_1"><span class="lab"><font color="Green" title="E60">A30</font></span></a>, <a class="txt" href="#E11:87_3_1"><span class="lab"><font color="Green" title="E61">A31</font></span></a>, <a class="ref" href="relat_1.html#D1" onmouseover="rs('relat_1/D1')" onmouseout="rh()">RELAT_1:def 1</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">y</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">  ex <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Olive" title="b1">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">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="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c19">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><br/>

<span class="kw">thus </span><a NAME="E14:87_3_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">y</font>,<font color="Maroon" title="c19">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_3_1"><span class="lab"><font color="Green" title="E62">A32</font></span></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>
<a NAME="E10:87_3"/>
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y1</font>, <font color="Olive" title="b3">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> holds <br/><font color="Olive" title="b2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">y2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y1</font>, <font color="Maroon" title="c12">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> implies <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</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"> (  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> or <font color="Maroon" title="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font> )</span><br/>

<a NAME="E2:87_3_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Y1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E4:87_3_2"/><span class="lab"><font color="Green" title="E59">A33</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y1</font> &amp; <font color="Maroon" title="c13">Y1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">p1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:87_3_2"/><span class="lab"><font color="Green" title="E60">A34</font></span>: 
( <font color="Maroon" title="c14">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c13">Y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3_2"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">p1</font> = <font color="Maroon" title="c14">p1</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_2"><span class="lab"><font color="Green" title="E60">A34</font></span></a></span>;<br/>
<a NAME="E8:87_3_2"/>
<font color="Maroon" title="c15">p1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_2"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E9:87_3_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p1</font> &amp; <font color="Maroon" title="c15">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m19</font>, <font color="Maroon" title="c17">m29</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c18">f19</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m19</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c19">f29</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m29</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E11:87_3_2"/><span class="lab"><font color="Green" title="E61">A35</font></span>: 
( <font color="Maroon" title="c16">m19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p1</font> &amp; <font color="Maroon" title="c17">m29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">p1</font> &amp; <font color="Maroon" title="c18">f19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m19</font> &amp; <font color="Maroon" title="c19">f29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m29</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f19</font>,<font color="Maroon" title="c19">f29</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/>
<a NAME="E12:87_3_2"/><span class="lab"><font color="Green" title="E62">A36</font></span>: 
<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> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f19</font>,<font color="Maroon" title="c19">f29</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3_2"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="txt" href="#E6:87_3_2"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E11:87_3_2"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E13:87_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">m19</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E14:87_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A37</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">m19</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E15:87_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</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="c11">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y2</font></span><br/>

<a NAME="E16:87_3_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">Y2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E18:87_3_2"/><span class="lab"><font color="Green" title="E64">A38</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Y2</font> &amp; <font color="Maroon" title="c20">Y2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c21">p2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E20:87_3_2"/><span class="lab"><font color="Green" title="E65">A39</font></span>: 
( <font color="Maroon" title="c21">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c20">Y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">p2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_3_2"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">p2</font> = <font color="Maroon" title="c21">p2</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:87_3_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a></span>;<br/>
<a NAME="E22:87_3_2"/>
<font color="Maroon" title="c22">p2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:87_3_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E23:87_3_2"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">p2</font> &amp; <font color="Maroon" title="c22">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">m199</font>, <font color="Maroon" title="c24">m299</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c25">f199</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c23">m199</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c26">f299</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c24">m299</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E25:87_3_2"/><span class="lab"><font color="Green" title="E66">A40</font></span>: 
( <font color="Maroon" title="c23">m199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p2</font> &amp; <font color="Maroon" title="c24">m299</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c22">p2</font> &amp; <font color="Maroon" title="c25">f199</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">m199</font> &amp; <font color="Maroon" title="c26">f299</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">m299</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c25">f199</font>,<font color="Maroon" title="c26">f299</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/>
<a NAME="E26:87_3_2"/><span class="lab"><font color="Green" title="E67">A41</font></span>: 
<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> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c25">f199</font>,<font color="Maroon" title="c26">f299</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_3_2"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E20:87_3_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E25:87_3_2"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E27:87_3_2"/><span class="kw">then </span>
<font color="Maroon" title="c10">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c23">m199</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E28:87_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A42</font></span>: 
( <font color="Maroon" title="c18">f19</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c25">f199</font> &amp; <font color="Maroon" title="c19">f29</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c26">f299</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:87_3_2"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="txt" href="#E25:87_3_2"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="txt" href="#E14:87_3_2"><span class="lab"><font color="Green" title="E63">A37</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E29:87_3_2"/><span class="lab"><font color="Green" title="E69">A43</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m19</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m29</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="#E12:87_3_2"><span class="lab"><font color="Green" title="E62">A36</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">x1</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m19</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m29</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">m19</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E12:87_3_2"><span class="lab"><font color="Green" title="E62">A36</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">y19</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m19</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:87_3_2"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c29">z1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m29</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:87_3_2"><span class="lab"><font color="Green" title="E69">A43</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E33:87_3_2"/><span class="lab"><font color="Green" title="E70">A44</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c23">m199</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c24">m299</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="#E26:87_3_2"><span class="lab"><font color="Green" title="E67">A41</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c30">x2</font> = <font color="Maroon" title="c10">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c23">m199</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c24">m299</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c23">m199</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E26:87_3_2"><span class="lab"><font color="Green" title="E67">A41</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">y29</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c23">m199</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:87_3_2"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">z2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c24">m299</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:87_3_2"><span class="lab"><font color="Green" title="E70">A44</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">thus </span><font color="Maroon" title="c11">y1</font> = 
<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f19</font>,<font color="Maroon" title="c19">f29</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_3_2"><span class="lab"><font color="Green" title="E59">A33</font></span></a>, <a class="txt" href="#E6:87_3_2"><span class="lab"><font color="Green" title="E60">A34</font></span></a>, <a class="txt" href="#E11:87_3_2"><span class="lab"><font color="Green" title="E61">A35</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">f19</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c28">y19</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">f29</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c29">z1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D20" target="_self" onmouseover="rs('algstr_4/D20')" onmouseout="rh()">Def20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c25">f199</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c31">y29</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">f299</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c32">z2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:87_3_2"><span class="lab"><font color="Green" title="E68">A42</font></span></a></span>
<br/>.= 
<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c25">f199</font>,<font color="Maroon" title="c26">f299</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">x2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D20" target="_self" onmouseover="rs('algstr_4/D20')" onmouseout="rh()">Def20</a></span>
<br/>.= 
<font color="Maroon" title="c12">y2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_3_2"><span class="lab"><font color="Green" title="E64">A38</font></span></a>, <a class="txt" href="#E20:87_3_2"><span class="lab"><font color="Green" title="E65">A39</font></span></a>, <a class="txt" href="#E25:87_3_2"><span class="lab"><font color="Green" title="E66">A40</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f9</font> = <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_3"><span class="lab"><font color="Green" title="E58">A28</font></span></a>, <a class="ref" href="funct_1.html#D1" onmouseover="rs('funct_1/D1')" onmouseout="rh()">FUNCT_1:def 1</a>, <a class="ref" href="relat_1.html#D1" onmouseover="rs('relat_1/D1')" onmouseout="rh()">RELAT_1:def 1</a></span>;<br/>
<a NAME="E12:87_3"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>) iff  ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>) iff  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font> )</span><br/>

<div><span class="kw">hereby </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"> (  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font> implies <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:87_3_3_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">p</font>, <font color="Maroon" title="c13">m</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:87_3_3_1"/><span class="lab"><font color="Green" title="E59">A45</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c12">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c12">p</font>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c13">m</font>) &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c11">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"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c12">p</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c13">m</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">p</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E1:87_3"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="ref" href="algstr_4.html#T21" target="_self" onmouseover="rs('algstr_4/T21')" onmouseout="rh()">Th21</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c14">m1</font>, <font color="Maroon" title="c15">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c16">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c14">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c17">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E5:87_3_3_1"/><span class="lab"><font color="Green" title="E60">A46</font></span>: 
( <font color="Maroon" title="c14">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c15">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">p</font> &amp; <font color="Maroon" title="c16">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">m1</font> &amp; <font color="Maroon" title="c17">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c17">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="txt" href="#E3:87_3_3_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">x9</font> = <font color="Maroon" title="c11">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c14">m1</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E3:87_3_3_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E5:87_3_3_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">y9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c14">m1</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_3_3_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E5:87_3_3_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">z9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_3_3_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E5:87_3_3_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c16">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">y9</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">z9</font></span>)</span> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  ;<br/><a NAME="E10:87_3_3_1"/><span class="lab"><font color="Green" title="E61">A47</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c17">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c14">m1</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c14">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/><a NAME="E11:87_3_3_1"/><span class="lab"><font color="Green" title="E62">A48</font></span>: 
<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c16">f1</font>,<font color="Maroon" title="c17">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D20" target="_self" onmouseover="rs('algstr_4/D20')" onmouseout="rh()">Def20</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">y</font> = <font color="Maroon" title="c21">y</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"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c22">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font></span><br/><a NAME="E12:87_3_3_1"/><span class="lab"><font color="Green" title="E63">A49</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c22">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:87_3_3_1"><span class="lab"><font color="Green" title="E60">A46</font></span></a>, <a class="txt" href="#E10:87_3_3_1"><span class="lab"><font color="Green" title="E61">A47</font></span></a>, <a class="txt" href="#E11:87_3_3_1"><span class="lab"><font color="Green" title="E62">A48</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/><a NAME="E13:87_3_3_1"/>
<font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_3_3_1"><span class="lab"><font color="Green" title="E59">A45</font></span></a>, <a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/><a NAME="E14:87_3_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/><a NAME="E15:87_3_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E16:87_3_3_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c22">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_3_3_1"><span class="lab"><font color="Green" title="E63">A49</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E17:87_3_3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c22">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</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">given </span><font color="Maroon" title="c12">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><a NAME="E3:87_3_3"/><span class="lab"><font color="Green" title="E59">A50</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f9</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="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span><br/>

<a NAME="E4:87_3_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E3:87_3_3"><span class="lab"><font color="Green" title="E59">A50</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E6:87_3_3"/><span class="lab"><font color="Green" title="E60">A51</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:87_3_3"/><span class="lab"><font color="Green" title="E61">A52</font></span>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_3"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E9:87_3_3"/><span class="lab"><font color="Green" title="E62">A53</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3_3"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c14">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3_3"><span class="lab"><font color="Green" title="E61">A52</font></span></a></span>;<br/>
<a NAME="E11:87_3_3"/>
( <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_3_3"><span class="lab"><font color="Green" title="E62">A53</font></span></a>, <a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m1</font>, <font color="Maroon" title="c17">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c18">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c19">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E13:87_3_3"/><span class="lab"><font color="Green" title="E63">A54</font></span>: 
( <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> &amp; <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f1</font>,<font color="Maroon" title="c19">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/>
<a NAME="E14:87_3_3"/><span class="lab"><font color="Green" title="E64">A55</font></span>: 
<font color="Maroon" title="c11">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> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f1</font>,<font color="Maroon" title="c19">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_3_3"><span class="lab"><font color="Green" title="E60">A51</font></span></a>, <a class="txt" href="#E8:87_3_3"><span class="lab"><font color="Green" title="E61">A52</font></span></a>, <a class="txt" href="#E13:87_3_3"><span class="lab"><font color="Green" title="E63">A54</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E15:87_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A56</font></span>: 
( <font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c11">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c16">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E16:87_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A57</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m1</font>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m2</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E17:87_3_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87_3_3"><span class="lab"><font color="Green" title="E64">A55</font></span></a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<a NAME="E18:87_3_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A58</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_3_3"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="mcart_1.html#T21" onmouseover="rs('mcart_1/T21')" onmouseout="rh()">MCART_1:21</a></span>;<br/>
<a NAME="E19:87_3_3"/>
<font color="Maroon" title="c11">x</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">m1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_3_3"><span class="lab"><font color="Green" title="E65">A56</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E20:87_3_3"/><span class="kw">then </span>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">m1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c17">m2</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87_3_3"><span class="lab"><font color="Green" title="E67">A58</font></span></a>, <a class="ref" href="algstr_4.html#T22" target="_self" onmouseover="rs('algstr_4/T22')" onmouseout="rh()">Th22</a>, <a class="txt" href="#E16:87_3_3"><span class="lab"><font color="Green" title="E66">A57</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:87_3_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_3_3"><span class="lab"><font color="Green" title="E63">A54</font></span></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>
<a NAME="E13:87_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A59</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c10">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xtuple_0.html#D12" onmouseover="rs('xtuple_0/D12')" onmouseout="rh()">XTUPLE_0:def 12</a></span>;<br/>
<a NAME="E14:87_3"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">f9</font> holds <br/><font color="Olive" title="b1">y</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 <font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_3_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">f9</font> implies <font color="Maroon" title="c11">y</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 <font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_3_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">f9</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="c11">y</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 <font color="Maroon" title="c2">M</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:87_3_4"/><span class="lab"><font color="Green" title="E60">A60</font></span>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c10">f9</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:87_3_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c9">fs1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:87_3"><span class="lab"><font color="Green" title="E57">A27</font></span></a>, <a class="txt" href="#E3:87_3_4"><span class="lab"><font color="Green" title="E60">A60</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E5:87_3_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E7:87_3_4"/><span class="lab"><font color="Green" title="E61">A61</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">Y</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:87_3_4"/><span class="lab"><font color="Green" title="E62">A62</font></span>: 
( <font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">fs1</font> &amp; <font color="Maroon" title="c13">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3_4"><span class="lab"><font color="Green" title="E61">A61</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E10:87_3_4"/><span class="lab"><font color="Green" title="E63">A63</font></span>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_3_4"><span class="lab"><font color="Green" title="E62">A62</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c14">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_3_4"><span class="lab"><font color="Green" title="E62">A62</font></span></a></span>;<br/>
<a NAME="E12:87_3_4"/>
( <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c15">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_3_4"><span class="lab"><font color="Green" title="E63">A63</font></span></a>, <a class="txt" href="#E6:87_3"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">m1</font>, <font color="Maroon" title="c17">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c18">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c16">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c19">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c17">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E14:87_3_4"/><span class="lab"><font color="Green" title="E64">A64</font></span>: 
( <font color="Maroon" title="c16">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c17">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c15">p</font> &amp; <font color="Maroon" title="c18">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font> &amp; <font color="Maroon" title="c19">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">m2</font> &amp; <font color="Maroon" title="c9">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f1</font>,<font color="Maroon" title="c19">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a></span>;<br/>
<a NAME="E15:87_3_4"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c18">f1</font>,<font color="Maroon" title="c19">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_3_4"><span class="lab"><font color="Green" title="E61">A61</font></span></a>, <a class="txt" href="#E9:87_3_4"><span class="lab"><font color="Green" title="E62">A62</font></span></a>, <a class="txt" href="#E14:87_3_4"><span class="lab"><font color="Green" title="E64">A64</font></span></a>, <a class="ref" href="xtuple_0.html#D13" onmouseover="rs('xtuple_0/D13')" onmouseout="rh()">XTUPLE_0:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:87_3_4"/>
<font color="Maroon" title="c11">y</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 <font color="Maroon" title="c2">M</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:87_3"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c10">f9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E16:87_3"/>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c7">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87_3"><span class="lab"><font color="Green" title="E59">A59</font></span></a>, <a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</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>
<a NAME="E15:87"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  holds <br/><font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a>  implies <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_4"/><span class="lab"><font color="Green" title="E54">A65</font></span>: 
<font color="Maroon" title="c7">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">fs</font> = <font color="Maroon" title="c7">e</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_4_1"/>
(  for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> holds <br/><font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> or  ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> &amp; <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_4_1_1"/><span class="lab"><font color="Green" title="E55">A66</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> holds <br/><font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/><div class="add"><a NAME="E2:87_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A67</font></span>: 
( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_4"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a></span>;<br/><a NAME="E3:87_4_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><a NAME="E4:87_4_1_1"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:87_4_1_1"/><span class="kw">then </span>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E6:87_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A68</font></span>: 
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_4_1_1_1"/>
(  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_4_1_1"><span class="lab"><font color="Green" title="E57">A68</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_4_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A69</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/><div class="add"><a NAME="E2:87_4_1_1_1_1"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T57" onmouseover="rs('funct_5/T57')" onmouseout="rh()">FUNCT_5:57</a></span>;<br/><a NAME="E3:87_4_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A70</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:87_4_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ,<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/><a NAME="E5:87_4_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E6:87_4_1_1_1_1"/><span class="kw">then </span>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E7:87_4_1_1_1_1"/><span class="kw">then </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_4_1_1_1_1"><span class="lab"><font color="Green" title="E59">A70</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:87_4_1_1_1_1"/>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_4_1_1_1_1"><span class="lab"><font color="Green" title="E58">A69</font></span></a>, <a class="txt" href="#E2:87_4_1_1"><span class="lab"><font color="Green" title="E56">A67</font></span></a>, <a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_4_1_1_1_2"/>
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:87_4_1_1_1_2"/>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E1:87_4_1_1"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="txt" href="#E1:87_4"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:87_4_1_1_1_3"/><span class="lab"><font color="Green" title="E58">A71</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">n</font> =  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font>;<br/><a NAME="E2:87_4_1_1_1_3"/>
 ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_4_1_1"><span class="lab"><font color="Green" title="E55">A66</font></span></a>, <a class="txt" href="#E1:87_4_1_1_1_3"><span class="lab"><font color="Green" title="E58">A71</font></span></a>, <a class="txt" href="#E1:87_4"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a></span>;<br/><a NAME="E3:87_4_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A72</font></span>: 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E1:87_4_1_1_1_3"><span class="lab"><font color="Green" title="E58">A71</font></span></a>, <a class="txt" href="#E1:87_4_1_1"><span class="lab"><font color="Green" title="E55">A66</font></span></a></span>;<br/><a NAME="E4:87_4_1_1_1_3"/><span class="lab"><font color="Green" title="E60">A73</font></span>: 
 <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E5:87_4_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon">S<sub>1</sub></font>[ <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font>,<font color="Maroon" title="c4">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E47">A2</font></span></a></span>;<br/><a NAME="E6:87_4_1_1_1_3"/><span class="kw">then </span>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font></span>)</span>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_4_1_1_1_3"><span class="lab"><font color="Green" title="E60">A73</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E7:87_4_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">F1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_4_1_1_1_3"><span class="lab"><font color="Green" title="E59">A72</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E8:87_4_1_1_1_3"/>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_4_1_2"/>
 ex <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c8">fs</font> &amp; <font color="Maroon" title="c8">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is  not   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</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="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:87_4_1_2"/>
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E1:87_4"><span class="lab"><font color="Green" title="E54">A65</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F2</font> = <font color="Maroon" title="c6">F2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a></span>)</span>,<font color="Maroon" title="c5">X1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="ref" href="funct_2.html#T3" onmouseover="rs('funct_2/T3')" onmouseout="rh()">FUNCT_2:3</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">X1</font> = <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">F3</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <font color="Maroon" title="c5">X1</font><span class="kw"> such that </span><br/><a NAME="E18:87"/><span class="lab"><font color="Green" title="E54">A74</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="algstr_4.html#S4" target="_self" onmouseover="rs('algstr_4/S4')" onmouseout="rh()">ALGSTR_4:sch 4</a>();<br/></span>
<a NAME="E19:87"/><span class="lab"><font color="Green" title="E55">A75</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>;<br/>
<a NAME="E1:87_5"/><span class="lab"><font color="Green" title="E56">A76</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_5_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">k</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_5_1"/><span class="lab"><font color="Green" title="E57">A77</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</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">S<sub>3</sub></font>[<font color="Maroon" title="c9">k</font>]</span><br/>

<span class="kw">thus </span><a NAME="E2:87_5_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c9">k</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_5_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_5_1_1"/><span class="lab"><font color="Green" title="E58">A78</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</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"> (  not <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:87_5_1_1"/>
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E3:87_5_1_1"/><span class="lab"><font color="Green" title="E59">A79</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c11">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c11">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_5_1_1_1"/><span class="lab"><font color="Green" title="E60">A80</font></span>: 
<font color="Maroon" title="c11">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</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="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c11">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E2:87_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A81</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c11">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<a NAME="E3:87_5_1_1_1"/>
<font color="Maroon" title="c11">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_5_1_1"><span class="lab"><font color="Green" title="E58">A78</font></span></a>, <a class="txt" href="#E1:87_5_1_1_1"><span class="lab"><font color="Green" title="E60">A80</font></span></a></span>;<br/>
<a NAME="E4:87_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:87_5_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c11">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_5_1_1_1"><span class="lab"><font color="Green" title="E61">A81</font></span></a>, <a class="txt" href="#E1:87_5_1"><span class="lab"><font color="Green" title="E57">A77</font></span></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>
<a NAME="E4:87_5_1_1"/><span class="lab"><font color="Green" title="E60">A82</font></span>: 
<font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">fs</font> = <font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">n</font> as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font> ;<br/>
<a NAME="E6:87_5_1_1"/><span class="lab"><font color="Green" title="E61">A83</font></span>: 
( (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) &amp; (  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> ) &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 &amp;  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> holds <br/> ex <font color="Olive" title="b2">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b3">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b3">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b4">m1</font>, <font color="Olive" title="b5">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b6">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b7">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b5">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b4">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b5">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">p</font> &amp; <font color="Olive" title="b6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m1</font> &amp; <font color="Olive" title="b7">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">m2</font> &amp; <font color="Olive" title="b2">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b6">f1</font>,<font color="Olive" title="b7">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b2">fs1</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_5_1_1"><span class="lab"><font color="Green" title="E59">A79</font></span></a>, <a class="txt" href="#E4:87_5_1_1"><span class="lab"><font color="Green" title="E60">A82</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a></span>;<br/>
<a NAME="E7:87_5_1_1"/><span class="lab"><font color="Green" title="E62">A84</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E8:87_5_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:87_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A85</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_5_1_1"><span class="lab"><font color="Green" title="E62">A84</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E64">A86</font></span>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c11">fs</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_5_1_1"><span class="lab"><font color="Green" title="E63">A85</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<a NAME="E11:87_5_1_1"/>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_5_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:87_5_1_1_3"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E2:87_5_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E3:87_5_1_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:87_5_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A87</font></span>: 
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_5_1_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_5_1_1_3_1"/>
( <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:87_5_1_1_3"><span class="lab"><font color="Green" title="E65">A87</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_5_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_5_1_1_3_1_1"/><span class="lab"><font color="Green" title="E66">A88</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:87_5_1_1_3_1_1"/>
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_5.html#T57" onmouseover="rs('funct_5/T57')" onmouseout="rh()">FUNCT_5:57</a></span>;<br/><a NAME="E3:87_5_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A88</font></span></a>, <a class="txt" href="#E6:87_5_1_1"><span class="lab"><font color="Green" title="E61">A83</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E4:87_5_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_5_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A88</font></span></a>, <a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/><a NAME="E5:87_5_1_1_3_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>) &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:87_5_1_1_3_1_1"/>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_5_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_5_1_1_3_1_2"/><span class="lab"><font color="Green" title="E66">A89</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:87_5_1_1_3_1_2"/>
 <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:87_5_1_1_3_1_2"/>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_5_1_1"><span class="lab"><font color="Green" title="E59">A79</font></span></a>, <a class="txt" href="#E4:87_5_1_1"><span class="lab"><font color="Green" title="E60">A82</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E1:87_5_1_1_3_1_2"><span class="lab"><font color="Green" title="E66">A89</font></span></a>, <a class="txt" href="#E10:87_5_1_1"><span class="lab"><font color="Green" title="E64">A86</font></span></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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_5_1_1_3_1_3"><span class="kw">suppose </span></a><a NAME="E1:87_5_1_1_3_1_3"/><span class="lab"><font color="Green" title="E66">A90</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 ; <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="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/><div class="add"><a NAME="E2:87_5_1_1_3_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; (  for <font color="Olive" title="b2">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b2">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b3">m1</font>, <font color="Olive" title="b4">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b5">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b6">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b4">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b3">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b4">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b2">p</font> &amp; <font color="Olive" title="b5">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m1</font> &amp; <font color="Olive" title="b6">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">fs</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">m2</font> &amp; <font color="Olive" title="b1">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b5">f1</font>,<font color="Olive" title="b6">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> ) ) &amp; <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Olive" title="b1">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_5_1_1"><span class="lab"><font color="Green" title="E59">A79</font></span></a>, <a class="txt" href="#E4:87_5_1_1"><span class="lab"><font color="Green" title="E60">A82</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E10:87_5_1_1"><span class="lab"><font color="Green" title="E64">A86</font></span></a></span>;<br/><a NAME="E3:87_5_1_1_3_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:87"><span class="lab"><font color="Green" title="E53">A23</font></span></a>, <a class="txt" href="#E1:87_5_1_1_3_1_3"><span class="lab"><font color="Green" title="E66">A90</font></span></a>, <a class="txt" href="#E10:87_5_1_1"><span class="lab"><font color="Green" title="E64">A86</font></span></a>, <a class="txt" href="#E3:87_5_1_1"><span class="lab"><font color="Green" title="E59">A79</font></span></a></span>;<br/><a NAME="E4:87_5_1_1_3_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> &amp;  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>) &amp;  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Olive" title="b1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D2" onmouseover="rs('funct_2/D2')" onmouseout="rh()">FUNCT_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E5:87_5_1_1_3_1_3"/>
<font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <font color="Maroon" title="c11">fs</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:87_5_1_1"/>
<font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c10">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c10">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87"><span class="lab"><font color="Green" title="E54">A74</font></span></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>


</div><span class="kw">end;</span></div>
<a NAME="E2:87_5"/>
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E1:87_5"><span class="lab"><font color="Green" title="E56">A76</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:87_5"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">n</font>)</span>)</span>,<font color="Maroon" title="c2">M</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c9">X9</font> =  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span><br/>  for <font color="Olive" title="b2">f9</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font>  st <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b2">f9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>;<br/>
<a NAME="E21:87"/><span class="lab"><font color="Green" title="E56">A91</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X9</font> holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</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 <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_6"><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">object</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X9</font> implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</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 <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_6"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X9</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">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</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 <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">w</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">f9</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">y</font> = <font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></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"> ( <font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <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 <font color="Maroon" title="c2">M</font> &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>] )</span><br/>

<a NAME="E4:87_6"/>
<font color="Maroon" title="c11">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="ref" href="algstr_4.html#T25" target="_self" onmouseover="rs('algstr_4/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E5:87_6"/><span class="kw">then </span>
<font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:87_6"/>
<font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</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">S<sub>3</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>]</span><br/>

<span class="kw">thus </span><a NAME="E7:87_6"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c12">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c10">h</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c9">X9</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E23:87"/><span class="lab"><font color="Green" title="E57">A92</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">X9</font> holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E21:87"><span class="lab"><font color="Green" title="E56">A91</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>,<font color="Maroon" title="c2">M</font> ;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c11">h</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="c11">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a>  &amp; <font color="Maroon" title="c11">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> )</span><br/>

<a NAME="E25:87"/>
 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="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> holds  <font color="Maroon" title="c11">h</font> <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"><font color="Maroon" title="c11">h</font> <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"><font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">b</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">a</font>, <font color="Maroon" title="c13">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="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</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"> <font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">fab</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<a NAME="E2:87_7"/>
<font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T31" target="_self" onmouseover="rs('algstr_4/T31')" onmouseout="rh()">Th31</a></span>;<br/>
<a NAME="E3:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A93</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> )
 
;<br/>
<a NAME="E4:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A94</font></span>: 
<font color="Maroon" title="c14">fab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F2</font> <a href="algstr_4.html#K2" title="ALGSTR_4:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87"><span class="lab"><font color="Green" title="E54">A74</font></span></a></span>;<br/>
<a NAME="E5:87_7"/><span class="lab"><font color="Green" title="E60">A95</font></span>: 
<font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<a NAME="E6:87_7"/><span class="lab"><font color="Green" title="E61">A96</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c15">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_7_1"/><span class="lab"><font color="Green" title="E62">A97</font></span>: 
<font color="Maroon" title="c15">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></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="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E2:87_7_1"/>
<font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c15">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:87_7_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c15">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_7_1"><span class="lab"><font color="Green" title="E62">A97</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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">set </span><font color="Maroon" title="c15">n</font> = <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span>;<br/>
<a NAME="E7:87_7"/>
(  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T32" target="_self" onmouseover="rs('algstr_4/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E8:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A98</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/>
<a NAME="E9:87_7"/><span class="lab"><font color="Green" title="E63">A99</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E10:87_7"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E11:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A100</font></span>: 
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:87_7"><span class="lab"><font color="Green" title="E63">A99</font></span></a>, <a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:87_7"><span class="lab"><font color="Green" title="E64">A100</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">fs1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E14:87_7"/><span class="lab"><font color="Green" title="E65">A101</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fs1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">and </span><br/><a NAME="E15:87_7"/><span class="lab"><font color="Green" title="E66">A102</font></span>: 
 for <font color="Olive" title="b1">p</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 holds <br/> ex <font color="Olive" title="b2">m1</font>, <font color="Olive" title="b3">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ex <font color="Olive" title="b4">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b2">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> ex <font color="Olive" title="b5">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b3">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> st <br/>( <font color="Olive" title="b2">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b3">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b1">p</font> &amp; <font color="Olive" title="b4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">m1</font> &amp; <font color="Olive" title="b5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">m2</font> &amp; <font color="Maroon" title="c16">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Olive" title="b4">f1</font>,<font color="Olive" title="b5">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">and </span><br/><a NAME="E16:87_7"/><span class="lab"><font color="Green" title="E67">A103</font></span>: 
<font color="Maroon" title="c14">fab</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c16">fs1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:87_7"><span class="lab"><font color="Green" title="E61">A96</font></span></a>, <a class="txt" href="#E8:87_7"><span class="lab"><font color="Green" title="E62">A98</font></span></a>, <a class="txt" href="#E5:87_7"><span class="lab"><font color="Green" title="E60">A95</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a>, <a class="txt" href="#E4:87_7"><span class="lab"><font color="Green" title="E59">A94</font></span></a></span>;<br/>
<a NAME="E17:87_7"/>
<font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="ref" href="algstr_4.html#T25" target="_self" onmouseover="rs('algstr_4/T25')" onmouseout="rh()">Th25</a></span>;<br/>
<a NAME="E18:87_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E19:87_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">fab</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:87_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">fab</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c16">fs1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:87_7"><span class="lab"><font color="Green" title="E67">A103</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E21:87_7"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">fab</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E23:87_7"/><span class="lab"><font color="Green" title="E68">A104</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">fab</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">Y</font> &amp; <font color="Maroon" title="c17">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c16">fs1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c18">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E25:87_7"/><span class="lab"><font color="Green" title="E69">A105</font></span>: 
( <font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">fs1</font> &amp; <font color="Maroon" title="c17">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87_7"><span class="lab"><font color="Green" title="E68">A104</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E26:87_7"/><span class="lab"><font color="Green" title="E70">A106</font></span>: 
<font color="Maroon" title="c18">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c16">fs1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:87_7"><span class="lab"><font color="Green" title="E69">A105</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">p</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:87_7"><span class="lab"><font color="Green" title="E69">A105</font></span></a></span>;<br/>
<a NAME="E28:87_7"/>
( <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c19">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:87_7"><span class="lab"><font color="Green" title="E70">A106</font></span></a>, <a class="txt" href="#E14:87_7"><span class="lab"><font color="Green" title="E65">A101</font></span></a>, <a class="ref" href="finseq_1.html#T1" onmouseover="rs('finseq_1/T1')" onmouseout="rh()">FINSEQ_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">m1</font>, <font color="Maroon" title="c21">m2</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>, <font color="Maroon" title="c22">f1</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>, <font color="Maroon" title="c23">f2</font> being   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</font>)</span>)</span>,<font color="Maroon" title="c2">M</font><span class="kw"> such that </span><br/><a NAME="E30:87_7"/><span class="lab"><font color="Green" title="E71">A107</font></span>: 
( <font color="Maroon" title="c20">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c21">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c22">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">m1</font> &amp; <font color="Maroon" title="c23">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">m2</font> &amp; <font color="Maroon" title="c16">fs1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c22">f1</font>,<font color="Maroon" title="c23">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_7"><span class="lab"><font color="Green" title="E66">A102</font></span></a></span>;<br/>
<a NAME="E31:87_7"/><span class="lab"><font color="Green" title="E72">A108</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c22">f1</font>,<font color="Maroon" title="c23">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a>, <a class="txt" href="#E23:87_7"><span class="lab"><font color="Green" title="E68">A104</font></span></a>, <a class="txt" href="#E25:87_7"><span class="lab"><font color="Green" title="E69">A105</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<a NAME="E32:87_7"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E33:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A109</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_7"><span class="lab"><font color="Green" title="E58">A93</font></span></a></span>;<br/>
<a NAME="E34:87_7"/><span class="kw">then </span>
<font color="Maroon" title="c20">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E35:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A110</font></span>: 
<font color="Maroon" title="c20">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E36:87_7"/>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T32" target="_self" onmouseover="rs('algstr_4/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E37:87_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E38:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A111</font></span>: 
<font color="Maroon" title="c20">m1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:87_7"><span class="lab"><font color="Green" title="E74">A110</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/>
<a NAME="E39:87_7"/>
 <a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#T32" target="_self" onmouseover="rs('algstr_4/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E40:87_7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E41:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A112</font></span>: 
<font color="Maroon" title="c21">m2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ordinal1.html#K6" title="ORDINAL1:func.6">Segm</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><a href="algstr_4.html#K15" title="ALGSTR_4:func.15">length</a> <font color="Maroon" title="c13">b</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:87_7"><span class="lab"><font color="Green" title="E74">A110</font></span></a>, <a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a>, <a class="ref" href="nat_1.html#T44" onmouseover="rs('nat_1/T44')" onmouseout="rh()">NAT_1:44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">x</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c20">m1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E31:87_7"><span class="lab"><font color="Green" title="E72">A108</font></span></a></span>;<br/>
<a NAME="E43:87_7"/><span class="lab"><font color="Green" title="E77">A113</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</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="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c20">m1</font>) <span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c24">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c21">m2</font>) <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:87_7"><span class="lab"><font color="Green" title="E77">A113</font></span></a>, <a class="ref" href="mcart_1.html#T10" onmouseover="rs('mcart_1/T10')" onmouseout="rh()">MCART_1:10</a></span>;<br/>
<a NAME="E46:87_7"/><span class="lab"><font color="Green" title="E78">A114</font></span>: 
<font color="Maroon" title="c24">x</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_7"><span class="lab"><font color="Green" title="E58">A93</font></span></a></span>;<br/>
<a NAME="E47:87_7"/><span class="lab"><font color="Green" title="E79">A115</font></span>: 
<span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c22">f1</font>,<font color="Maroon" title="c23">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">z</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D20" target="_self" onmouseover="rs('algstr_4/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<a NAME="E48:87_7"/><span class="lab"><font color="Green" title="E80">A116</font></span>: 
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">fab</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87"><span class="lab"><font color="Green" title="E57">A92</font></span></a></span>;<br/>
<a NAME="E49:87_7"/><span class="lab"><font color="Green" title="E81">A117</font></span>: 
<font color="Maroon" title="c14">fab</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">[:</a><span class="default"><font color="Maroon" title="c22">f1</font>,<font color="Maroon" title="c23">f2</font></span><a href="algstr_4.html#K17" title="ALGSTR_4:func.17">:]</a></span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a>, <a class="txt" href="#E23:87_7"><span class="lab"><font color="Green" title="E68">A104</font></span></a>, <a class="txt" href="#E25:87_7"><span class="lab"><font color="Green" title="E69">A105</font></span></a>, <a class="ref" href="funct_1.html#T1" onmouseover="rs('funct_1/T1')" onmouseout="rh()">FUNCT_1:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">fa</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c28">fb</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<font color="Maroon" title="c22">f1</font> = 
<font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">m1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a>, <a class="txt" href="#E38:87_7"><span class="lab"><font color="Green" title="E75">A111</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c27">fa</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:87_7"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>
;<br/>
<a NAME="E53:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A118</font></span>: 
<font color="Maroon" title="c27">fa</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">f1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:87_7"><span class="lab"><font color="Green" title="E78">A114</font></span></a></span>;<br/>
<font color="Maroon" title="c23">f2</font> = 
<font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c21">m2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a>, <a class="txt" href="#E41:87_7"><span class="lab"><font color="Green" title="E76">A112</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
<font color="Maroon" title="c28">fb</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D18" target="_self" onmouseover="rs('algstr_4/D18')" onmouseout="rh()">Def18</a>, <a class="txt" href="#E35:87_7"><span class="lab"><font color="Green" title="E74">A110</font></span></a>, <a class="txt" href="#E30:87_7"><span class="lab"><font color="Green" title="E71">A107</font></span></a></span>
;<br/>
<a NAME="E55:87_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A119</font></span>: 
<font color="Maroon" title="c28">fb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">f2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c26">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:87_7"><span class="lab"><font color="Green" title="E78">A114</font></span></a></span>;<br/>
<a NAME="E56:87_7"/>
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">fb</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">b</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87"><span class="lab"><font color="Green" title="E57">A92</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E57:87_7"/>
<font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:87_7"><span class="lab"><font color="Green" title="E79">A115</font></span></a>, <a class="txt" href="#E48:87_7"><span class="lab"><font color="Green" title="E80">A116</font></span></a>, <a class="txt" href="#E53:87_7"><span class="lab"><font color="Green" title="E82">A118</font></span></a>, <a class="txt" href="#E55:87_7"><span class="lab"><font color="Green" title="E83">A119</font></span></a>, <a class="txt" href="#E23:87"><span class="lab"><font color="Green" title="E57">A92</font></span></a>, <a class="txt" href="#E49:87_7"><span class="lab"><font color="Green" title="E81">A117</font></span></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="E26:87"/>
<font color="Maroon" title="c11">h</font> is  <a href="group_6.html#V1" title="GROUP_6:attr.1">multiplicative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</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="c11">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">fX</font> =  <a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1);<br/>
<a NAME="E27:87"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> holds <br/><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> <font color="Maroon" title="c11">h</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> implies <font color="Maroon" title="c13">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> <font color="Maroon" title="c11">h</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_8"/><span class="lab"><font color="Green" title="E58">A120</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></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"> <font color="Maroon" title="c13">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> <font color="Maroon" title="c11">h</font></span><br/>

<a NAME="E2:87_8"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" onmouseover="rs('relat_1/T25')" onmouseout="rh()">RELAT_1:25</a></span>;<br/>
<a NAME="E3:87_8"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_8"><span class="lab"><font color="Green" title="E58">A120</font></span></a></span>;<br/>
<a NAME="E4:87_8"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<a NAME="E5:87_8"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:87_8"/>
<font color="Maroon" title="c13">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> <font color="Maroon" title="c11">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</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>
<a NAME="E28:87"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A121</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h</font>
 
;<br/>
<a NAME="E29:87"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c11">h</font> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span></span>)</span> implies <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_9"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span></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"> <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span><br/>

<a NAME="E2:87_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A122</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:87"><span class="lab"><font color="Green" title="E58">A121</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>;<br/>
<a NAME="E3:87_9"/><span class="lab"><font color="Green" title="E60">A123</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T25" onmouseover="rs('relat_1/T25')" onmouseout="rh()">RELAT_1:25</a></span>;<br/>
<a NAME="E4:87_9"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_9"><span class="lab"><font color="Green" title="E59">A122</font></span></a></span>;<br/>
<a NAME="E5:87_9"/><span class="kw">then </span>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T33" onmouseover="rs('funct_1/T33')" onmouseout="rh()">FUNCT_1:33</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x9</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E7:87_9"/><span class="lab"><font color="Green" title="E61">A124</font></span>: 
( <font color="Maroon" title="c14">x9</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> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> &amp; <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E8:87_9"/><span class="lab"><font color="Green" title="E62">A125</font></span>: 
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:87_9"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="algstr_4.html#E39"><span class="lab"><font color="Green" title="E28">Lm1</font></span></a></span>;<br/>
<a NAME="E10:87_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A126</font></span>: 
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">X</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="algstr_4.html#K12" title="ALGSTR_4:func.12">free_magma_carrier</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="algstr_4.html#D13" target="_self" onmouseover="rs('algstr_4/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<a NAME="E11:87_9"/><span class="lab"><font color="Green" title="E64">A127</font></span>: 
<font color="Maroon" title="c14">x9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_9"><span class="lab"><font color="Green" title="E61">A124</font></span></a>, <a class="txt" href="algstr_4.html#E57"><span class="lab"><font color="Green" title="E41">Lm3</font></span></a></span>;<br/>
<a NAME="E12:87_9"/><span class="lab"><font color="Green" title="E65">A128</font></span>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">x9</font>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_9"><span class="lab"><font color="Green" title="E61">A124</font></span></a>, <a class="ref" href="algstr_4.html#D19" target="_self" onmouseover="rs('algstr_4/D19')" onmouseout="rh()">Def19</a></span>;<br/>
<a NAME="E13:87_9"/><span class="kw">then </span>
<font color="Maroon" title="c13">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"><font color="Maroon" title="c1">X</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="#E8:87_9"><span class="lab"><font color="Green" title="E62">A125</font></span></a>, <a class="txt" href="#E11:87_9"><span class="lab"><font color="Green" title="E64">A127</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">w</font> = <font color="Maroon" title="c13">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K14" title="ALGSTR_4:func.14">free_magma</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:87_9"><span class="lab"><font color="Green" title="E63">A126</font></span></a></span>;<br/>
<a NAME="E15:87_9"/><span class="lab"><font color="Green" title="E66">A129</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:87_9"><span class="lab"><font color="Green" title="E61">A124</font></span></a>, <a class="ref" href="funct_1.html#T34" onmouseover="rs('funct_1/T34')" onmouseout="rh()">FUNCT_1:34</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">f9</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c17">f9</font> = <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c15">w</font> <a href="xfamily.html#K2" title="XFAMILY:func.2">`2</a></span>)</span>)</span>)</span>,<font color="Maroon" title="c2">M</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E67">A130</font></span>: <font color="Maroon" title="c17">f9</font> = 
<font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_9"><span class="lab"><font color="Green" title="E65">A128</font></span></a></span>
<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:87"><span class="lab"><font color="Green" title="E54">A74</font></span></a></span>
;<br/>
<a NAME="E18:87_9"/><span class="lab"><font color="Green" title="E68">A131</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="87_9_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">m</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c18">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c18">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_9_2"/>
<font color="Maroon" title="c18">m</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</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="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c18">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font></span><br/>

<a NAME="E2:87_9_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">F3</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c18">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:87_9_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">m</font> is   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <span class="p1">(<span class="default"><a href="algstr_4.html#K11" title="ALGSTR_4:func.11">free_magma</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c18">m</font>)</span>)</span>,<font color="Maroon" title="c2">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:87"><span class="lab"><font color="Green" title="E55">A75</font></span></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>
<a NAME="E19:87_9"/><span class="lab"><font color="Green" title="E69">A132</font></span>: 
<font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">X1</font> <a href="afinsq_1.html#K8" title="AFINSQ_1:func.8">^omega</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#D7" onmouseover="rs('afinsq_1/D7')" onmouseout="rh()">AFINSQ_1:def 7</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">fs</font> = <font color="Maroon" title="c8">F3</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> 1 as   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of <font color="Maroon" title="c5">X1</font> ;<br/>
<a NAME="E21:87_9"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E22:87_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A133</font></span>: 
1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D2" onmouseover="rs('ordinal1/D2')" onmouseout="rh()">ORDINAL1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A134</font></span>:  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c18">fs</font> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">F3</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:87_9"><span class="lab"><font color="Green" title="E70">A133</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font> = 
<font color="Maroon" title="c17">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">w</font> <a href="xfamily.html#K1" title="XFAMILY:func.1">`1</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:87"><span class="lab"><font color="Green" title="E57">A92</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">f9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:87_9"><span class="lab"><font color="Green" title="E65">A128</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:87_9"><span class="lab"><font color="Green" title="E66">A129</font></span></a>, <a class="txt" href="#E17:87_9"><span class="lab"><font color="Green" title="E67">A130</font></span></a>, <a class="txt" href="#E23:87_9"><span class="lab"><font color="Green" title="E71">A134</font></span></a>, <a class="txt" href="#E18:87_9"><span class="lab"><font color="Green" title="E68">A131</font></span></a>, <a class="txt" href="#E19:87_9"><span class="lab"><font color="Green" title="E69">A132</font></span></a>, <a class="txt" href="#E13:87"><span class="lab"><font color="Green" title="E52">A22</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:87_9"><span class="lab"><font color="Green" title="E60">A123</font></span></a>, <a class="txt" href="#E2:87_9"><span class="lab"><font color="Green" title="E59">A122</font></span></a>, <a class="ref" href="funct_1.html#T13" onmouseover="rs('funct_1/T13')" onmouseout="rh()">FUNCT_1:13</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>
<a NAME="E30:87"/><span class="kw">then </span>
<font color="Maroon" title="c11">h</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" onmouseover="rs('partfun1/D4')" onmouseout="rh()">PARTFUN1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E31:87"/>
<font color="Maroon" title="c11">h</font> <a href="algstr_4.html#R1" title="ALGSTR_4:pred.1">extends</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_4.html#K16" title="ALGSTR_4:func.16">canon_image</a> (<font color="Maroon" title="c1">X</font>,1)</span>)</span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:87"><span class="lab"><font color="Green" title="E58">A121</font></span></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>
