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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">G</font> being   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>  st  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/> ex <font color="Olive" title="b2">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b3">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b2">I</font> ex <font color="Olive" title="b4">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b3">F</font></span>)</span>,<font color="Olive" title="b1">G</font> st <br/>( <font color="Olive" title="b2">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">I</font> holds <br/> ( <font color="Olive" title="b3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Olive" title="b1">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Olive" title="b1">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> ) ) &amp; (  for <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">I</font>  st <font color="Olive" title="b5">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b6">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b4">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b5">x</font> being  <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b6">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b2">I</font> holds  <font color="Olive" title="b5">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b6">p</font> ) holds <br/>( <font color="Olive" title="b5">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b4">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b5">x</font> ) ) );<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E2:42"/><span class="lab"><font color="Green" title="E44">A2</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">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">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">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>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">thus </span><a NAME="E2:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">G</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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"> (  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1"/><span class="lab"><font color="Green" title="E46">A4</font></span>: 
(  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_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="E1:42_1_1_1"/>
( <font color="Maroon" title="c1">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="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
<font color="Maroon" title="c1">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">  ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span>;<br/><a NAME="E2:42_1_1_1_1"/><span class="lab"><font color="Green" title="E48">A6</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E3:42_1_1_1_1"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:42_1_1_1_1"/><span class="lab"><font color="Green" title="E49">A7</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">q</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_1"><span class="lab"><font color="Green" title="E48">A6</font></span></a>, <a class="txt" href="#E5:42_1_1_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="nat_3.html#T34" onmouseover="rs('nat_3/T34')" onmouseout="rh()">NAT_3:34</a></span>;<br/><a NAME="E7:42_1_1_1_1"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">q0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E9:42_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A8</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <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"><font color="Maroon" title="c6">q0</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="card_1.html#T29" onmouseover="rs('card_1/T29')" onmouseout="rh()">CARD_1:29</a>, <a class="txt" href="#E1:42_1_1_1_1"><span class="lab"><font color="Green" title="E47">A5</font></span></a>, <a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a></span>;<br/><a NAME="E10:42_1_1_1_1"/><span class="lab"><font color="Green" title="E51">A9</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1_1_1_1"><span class="lab"><font color="Green" title="E50">A8</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E5:42_1_1_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">Gq</font> = <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c8">g</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font>;<br/><a NAME="E12:42_1_1_1_1"/><span class="lab"><font color="Green" title="E52">A10</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Gq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>;<br/><a NAME="E13:42_1_1_1_1"/>
<font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_1_1"/>
<font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span> <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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T55" onmouseover="rs('nat_3/T55')" onmouseout="rh()">NAT_3:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</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="E14:42_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A11</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <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"><font color="Maroon" title="c5">q</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="nat_3.html#T42" onmouseover="rs('nat_3/T42')" onmouseout="rh()">NAT_3:42</a></span>;<br/><a NAME="E15:42_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c7">Gq</font></span>)</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="E16:42_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A12</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c7">Gq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E17:42_1_1_1_1"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <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"><font color="Maroon" title="c5">q</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="#E14:42_1_1_1_1"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E18:42_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A13</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E19:42_1_1_1_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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <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> <font color="Olive" title="b1">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">p</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">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_1_2"/>
<font color="Maroon" title="c9">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:42_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <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="c5">q</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="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a>, <a class="txt" href="#E10:42_1_1_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a></span>;<br/>
<a NAME="E3:42_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">hence </span><a NAME="E4:42_1_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c7">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42_1_1_1_1"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="nat_3.html#T25" onmouseover="rs('nat_3/T25')" onmouseout="rh()">NAT_3:25</a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2: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">then </span><span class="lab"><font color="Green" title="E56">A14</font></span>: <font color="Maroon" title="c7">Gq</font> = 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:42_1_1_1_1"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a>, <a class="txt" href="#E12:42_1_1_1_1"><span class="lab"><font color="Green" title="E52">A10</font></span></a></span>
<br/>.= 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">I</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><span class="kw">set </span><font color="Maroon" title="c10">F</font> = <font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">G</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c2">G</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c9">I</font> ;<br/><a NAME="E23:42_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A16</font></span>: 
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">t</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">s</font>, <font color="Maroon" title="c13">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</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="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">t</font> implies  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<a NAME="E1:42_1_1_1_1_4"/>
<font color="Maroon" title="c12">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">hence </span><a NAME="E2:42_1_1_1_1_4"/>
( <font color="Maroon" title="c12">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">t</font> implies  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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>; <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="E24:42_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A18</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds <br/> ( <font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></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="42_1_1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</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">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )</span><br/>

<a NAME="E1:42_1_1_1_1_5"/>
<font color="Maroon" title="c12">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</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>, <a class="txt" href="#E5:42_1_1_1_1"><span class="lab"><font color="Green" title="E49">A7</font></span></a></span>;<br/>
<a NAME="E2:42_1_1_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A20</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<a NAME="E3:42_1_1_1_1_5"/><span class="lab"><font color="Green" title="E60">A21</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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 href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> = 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <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="txt" href="#E3:42_1_1_1_1_5"><span class="lab"><font color="Green" title="E60">A21</font></span></a>, <a class="txt" href="#E2:42_1_1_1_1_5"><span class="lab"><font color="Green" title="E59">A20</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a>, <a class="txt" href="#E20:42_1_1_1_1"><span class="lab"><font color="Green" title="E56">A14</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:42_1_1_1_1_5"/>
( <font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <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="group_2.html#T54" onmouseover="rs('group_2/T54')" onmouseout="rh()">GROUP_2:54</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">reconsider </span><font color="Maroon" title="c12">F</font> = <font color="Maroon" title="c11">F</font> as   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c9">I</font> ;<br/><span class="kw">take </span>
<font color="Maroon" title="c9">I</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">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c9">I</font> ex <font color="Olive" title="b2">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b1">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b3">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> ) ) &amp; (  for <font color="Olive" title="b3">p</font>, <font color="Olive" title="b4">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font>  st <font color="Olive" title="b3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b2">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b3">x</font> being  <font color="Maroon" title="c9">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds  <font color="Olive" title="b3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> ) holds <br/>( <font color="Olive" title="b3">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b2">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x</font> ) ) )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c12">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">  ex <font color="Olive" title="b1">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds <br/> ( <font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> ) ) &amp; (  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b1">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">x</font> being  <font color="Maroon" title="c9">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b3">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds  <font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> ) holds <br/>( <font color="Olive" title="b2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Olive" title="b1">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b2">x</font> ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font></span>)</span>,<font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E27:42_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A24</font></span>: 
( <font color="Maroon" title="c13">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b1">x</font> being  <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> holds  <font color="Maroon" title="c13">HFG</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>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T26" target="_self" onmouseover="rs('group_17/T26')" onmouseout="rh()">Th26</a></span>;<br/><a NAME="E28:42_1_1_1_1"/>
 for <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c9">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds  <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> ) holds <br/>( <font color="Olive" title="b1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Maroon" title="c13">HFG</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>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:42_1_1_1_1"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="ref" href="group_17.html#T25" target="_self" onmouseover="rs('group_17/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E29:42_1_1_1_1"/>
 ex <font color="Olive" title="b1">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Maroon" title="c9">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds <br/> ( <font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> ) ) &amp; (  for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font>  st <font color="Olive" title="b2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b1">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">x</font> being  <font color="Maroon" title="c9">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b3">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">I</font> holds  <font color="Olive" title="b2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> ) holds <br/>( <font color="Olive" title="b2">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c12">F</font> &amp; <font color="Olive" title="b1">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b2">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1_1_1_1"><span class="lab"><font color="Green" title="E51">A9</font></span></a>, <a class="txt" href="#E27:42_1_1_1_1"><span class="lab"><font color="Green" title="E59">A24</font></span></a>, <a class="txt" href="#E23:42_1_1_1_1"><span class="lab"><font color="Green" title="E57">A16</font></span></a>, <a class="txt" href="#E24:42_1_1_1_1"><span class="lab"><font color="Green" title="E58">A18</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="42_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2"/><span class="lab"><font color="Green" title="E47">A25</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&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">  ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c3">f</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span>;<br/><a NAME="E2:42_1_1_1_2"/><span class="lab"><font color="Green" title="E48">A26</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>;<br/><a NAME="E3:42_1_1_1_2"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E4:42_1_1_1_2"/>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">q</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:42_1_1_1_2"/><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c4">q</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="nat_3.html#T34" onmouseover="rs('nat_3/T34')" onmouseout="rh()">NAT_3:34</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">Gq</font> = <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> as   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/><span class="kw">set </span><font color="Maroon" title="c7">g</font> =  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">h</font> = <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span>;<br/><a NAME="E9:42_1_1_1_2"/>
<font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_1"/>
<font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span> <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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T55" onmouseover="rs('nat_3/T55')" onmouseout="rh()">NAT_3:55</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</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="E10:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A29</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <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"><font color="Maroon" title="c5">q</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="nat_3.html#T42" onmouseover="rs('nat_3/T42')" onmouseout="rh()">NAT_3:42</a></span>;<br/><a NAME="E11:42_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <font color="Maroon" title="c6">Gq</font></span>)</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="E12:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <font color="Maroon" title="c6">Gq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E13:42_1_1_1_2"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T25" onmouseover="rs('nat_3/T25')" onmouseout="rh()">NAT_3:25</a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a>, <a class="txt" href="#E12:42_1_1_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a></span>;<br/><a NAME="E14:42_1_1_1_2"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <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"><font color="Maroon" title="c5">q</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="#E10:42_1_1_1_2"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E15:42_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A33</font></span>: 
 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="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> iff <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="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> iff <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</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"> ( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:42_1_1_1_2_2_1"/><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><a NAME="E2:42_1_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T39" onmouseover="rs('pre_poly/T39')" onmouseout="rh()">PRE_POLY:39</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><a NAME="E3:42_1_1_1_2_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E4:42_1_1_1_2_2_1"/>
 not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A37</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="txt" href="#E3:42_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T25" onmouseover="rs('nat_3/T25')" onmouseout="rh()">NAT_3:25</a>, <a class="ref" href="int_2.html#D4" onmouseover="rs('int_2/D4')" onmouseout="rh()">INT_2:def 4</a>, <a class="txt" href="#E12:42_1_1_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span>
<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 href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/>
<span class="kw">hence </span><a NAME="E4:42_1_1_1_2_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:42_1_1_1_2_2_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E2:42_1_1_1_2_2_1"><span class="lab"><font color="Green" title="E57">A35</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">assume </span><a NAME="E2:42_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span></span><br/>

<a NAME="E3:42_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A38</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E4:42_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_2"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">x0</font> = <font color="Maroon" title="c9">x</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T34" onmouseover="rs('nat_3/T34')" onmouseout="rh()">NAT_3:34</a></span>;<br/>
<a NAME="E6:42_1_1_1_2_2"/><span class="lab"><font color="Green" title="E57">A39</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>;<br/>
<a NAME="E7:42_1_1_1_2_2"/><span class="lab"><font color="Green" title="E58">A40</font></span>: 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E3:42_1_1_1_2_2"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E8:42_1_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_2"><span class="lab"><font color="Green" title="E56">A38</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E9:42_1_1_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1_2_2"><span class="lab"><font color="Green" title="E57">A39</font></span></a>, <a class="txt" href="#E7:42_1_1_1_2_2"><span class="lab"><font color="Green" title="E58">A40</font></span></a>, <a class="ref" href="nat_d.html#T40" onmouseover="rs('nat_d/T40')" onmouseout="rh()">NAT_D:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:42_1_1_1_2_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</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="E16:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A41</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>;<br/><a NAME="E17:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A42</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">h</font> = <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="pre_poly.html#K12" title="PRE_POLY:func.12">-'</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> as   <a href="pre_poly.html#NM2" title="PRE_POLY:NM.2">bag</a> of  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a>  ;<br/><a NAME="E19:42_1_1_1_2"/><span class="lab"><font color="Green" title="E58">A43</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">n</font> &amp; <font color="Maroon" title="c9">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> <font color="Olive" title="b1">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b2">n</font> )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_3"/>
<font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">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">  ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )</span><br/>

<a NAME="E2:42_1_1_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A44</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c10">x</font> <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="c5">q</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a></span>;<br/>
<a NAME="E3:42_1_1_1_2_3"/><span class="lab"><font color="Green" title="E60">A45</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>;<br/>
<a NAME="E4:42_1_1_1_2_3"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_3"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/>
<a NAME="E5:42_1_1_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_3"><span class="lab"><font color="Green" title="E60">A45</font></span></a>, <a class="ref" href="nat_d.html#T40" onmouseover="rs('nat_d/T40')" onmouseout="rh()">NAT_D:40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:42_1_1_1_2_3"/>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> &amp; <font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <font color="Olive" title="b1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_3"><span class="lab"><font color="Green" title="E59">A44</font></span></a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7: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="E20:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c9">h</font> is  <a href="int_7.html#V1" title="INT_7:attr.1">prime-factorization-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a></span>;<br/><a NAME="E21:42_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><a NAME="E22:42_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A48</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" onmouseover="rs('xboole_1/T36')" onmouseout="rh()">XBOOLE_1:36</a>, <a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a></span>;<br/><a NAME="E23:42_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A49</font></span>: 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a>, <a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="xboole_1.html#T45" onmouseover="rs('xboole_1/T45')" onmouseout="rh()">XBOOLE_1:45</a></span>;<br/><a NAME="E24:42_1_1_1_2"/>
 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="newton.html#K8" title="NEWTON:func.8">SetPrimes</a>  holds <br/><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_4"><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>  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</a>  implies <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_4"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="newton.html#K8" title="NEWTON:func.8">SetPrimes</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">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_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="E1:42_1_1_1_2_4_1"/>
(  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E63">A50</font></span>: 
 not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A51</font></span>: 
 not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:42_1_1_1_2"><span class="lab"><font color="Green" title="E61">A48</font></span></a></span>;<br/><a NAME="E3:42_1_1_1_2_4_1_1"/><span class="lab"><font color="Green" title="E65">A52</font></span>: 
 not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E21:42_1_1_1_2"><span class="lab"><font color="Green" title="E60">A47</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> = 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a>, <a class="txt" href="#E1:42_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E63">A50</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a>, <a class="txt" href="#E2:42_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A51</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a>, <a class="txt" href="#E3:42_1_1_1_2_4_1_1"><span class="lab"><font color="Green" title="E65">A52</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="42_1_1_1_2_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E63">A53</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></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"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_4_1_2"/><span class="lab"><font color="Green" title="E64">A54</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><span class="kw">thus </span><a NAME="E3:42_1_1_1_2_4_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_4_1_2_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="E1:42_1_1_1_2_4_1_2_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E63">A53</font></span></a>, <a class="txt" href="#E23:42_1_1_1_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_4_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_4_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">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"> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_4_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A55</font></span>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:42_1_1_1_2_4_1_2_1_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">x0</font> = <font color="Maroon" title="c10">x</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T34" onmouseover="rs('nat_3/T34')" onmouseout="rh()">NAT_3:34</a></span>;<br/><a NAME="E5:42_1_1_1_2_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E66">A56</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x0</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x0</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>;<br/><a NAME="E6:42_1_1_1_2_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E2:42_1_1_1_2_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E65">A55</font></span></a>, <a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_1_1_1_2_4_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_2_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E66">A56</font></span></a>, <a class="ref" href="nat_d.html#T40" onmouseover="rs('nat_d/T40')" onmouseout="rh()">NAT_D:40</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="42_1_1_1_2_4_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_4_1_2_1_1_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_4_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A57</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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="E3:42_1_1_1_2_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E66">A58</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>;<br/><a NAME="E4:42_1_1_1_2_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> <a href="newton.html#K1" title="NEWTON:func.1">|^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="nat_3.html#K11" title="NAT_3:func.11">|-count</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A57</font></span></a>, <a class="txt" href="#E2:42_1_1_1_2_4_1_2"><span class="lab"><font color="Green" title="E64">A54</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E5:42_1_1_1_2_4_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">q</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E66">A58</font></span></a>, <a class="ref" href="xreal_0.html#D2" onmouseover="rs('xreal_0/D2')" onmouseout="rh()">XREAL_0:def 2</a>, <a class="txt" href="#E13:42_1_1_1_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:42_1_1_1_2_4_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E65">A57</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E25:42_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">h</font> <a href="pre_poly.html#K11" title="PRE_POLY:func.11">+</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a>  <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#T33" onmouseover="rs('pre_poly/T33')" onmouseout="rh()">PRE_POLY:33</a></span>;<br/><a NAME="E26:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A59</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">h</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="xboole_1.html#T79" onmouseover="rs('xboole_1/T79')" onmouseout="rh()">XBOOLE_1:79</a>, <a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="nat_3.html#T19" onmouseover="rs('nat_3/T19')" onmouseout="rh()">NAT_3:19</a></span>;<br/><a NAME="E27:42_1_1_1_2"/>
(  not  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">h</font> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  &amp;  not  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> is  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:42_1_1_1_2"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">H</font>, <font color="Maroon" title="c11">K</font> being   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E29:42_1_1_1_2"/><span class="lab"><font color="Green" title="E64">A60</font></span>: 
(  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c10">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">h</font> &amp;  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c11">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c10">H</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c11">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp;  ex <font color="Olive" title="b1">F</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c10">H</font>,<font color="Maroon" title="c11">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c10">H</font> &amp; <font color="Olive" title="b3">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c11">K</font> holds <br/><font color="Olive" title="b1">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">b</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2"><span class="lab"><font color="Green" title="E48">A26</font></span></a>, <a class="txt" href="#E26:42_1_1_1_2"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="ref" href="group_17.html#T18" target="_self" onmouseover="rs('group_17/T18')" onmouseout="rh()">Th18</a>, <a class="txt" href="#E20:42_1_1_1_2"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E17:42_1_1_1_2"><span class="lab"><font color="Green" title="E57">A42</font></span></a>, <a class="ref" href="int_7.html#T12" onmouseover="rs('int_7/T12')" onmouseout="rh()">INT_7:12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c12">H</font> = <font color="Maroon" title="c10">H</font>, <font color="Maroon" title="c13">K</font> = <font color="Maroon" title="c11">K</font> as   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a> ;<br/><a NAME="E31:42_1_1_1_2"/><span class="lab"><font color="Green" title="E65">A61</font></span>: 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a></span>;<br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span></span>)</span> = 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p3">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p4">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="euler_1.html#T4" onmouseover="rs('euler_1/T4')" onmouseout="rh()">EULER_1:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</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> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1"><span class="lab"><font color="Green" title="E46">A4</font></span></a>, <a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>

;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">I0</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> , <font color="Maroon" title="c15">F0</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c14">I0</font>, <font color="Maroon" title="c16">HFG0</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c15">F0</font></span>)</span>,<font color="Maroon" title="c12">H</font><span class="kw"> such that </span><br/><a NAME="E34:42_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A62</font></span>: 
( <font color="Maroon" title="c14">I0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> holds <br/> ( <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c12">H</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> ) ) &amp; (  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c12">H</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Maroon" title="c16">HFG0</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c14">I0</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> holds  <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> ) holds <br/>( <font color="Olive" title="b1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c15">F0</font> &amp; <font color="Maroon" title="c16">HFG0</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>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2"><span class="lab"><font color="Green" title="E47">A25</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">I</font> = <font color="Maroon" title="c14">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c18">I</font> = <font color="Maroon" title="c14">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ;<br/><a NAME="E36:42_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A63</font></span>: 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>;<br/><a NAME="E37:42_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A64</font></span>: 
 <a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p1">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c9">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:42_1_1_1_2"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a></span>;<br/><a NAME="E38:42_1_1_1_2"/><span class="lab"><font color="Green" title="E69">A65</font></span>: 
<font color="Maroon" title="c18">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E23:42_1_1_1_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E20:42_1_1_1_2"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a>, <a class="txt" href="#E36:42_1_1_1_2"><span class="lab"><font color="Green" title="E67">A63</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">F</font> = <font color="Maroon" title="c15">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>;<br/><span class="lab"><font color="Green" title="E70">A67</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="c15">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c14">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c14">I0</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">F</font> = <font color="Maroon" title="c15">F0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span> as  <font color="Maroon" title="c18">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D18" onmouseover="rs('relat_1/D18')" onmouseout="rh()">RELAT_1:def 18</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">F</font> = <font color="Maroon" title="c20">F</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <font color="Maroon" title="c18">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a>, <a class="txt" href="#E39:42_1_1_1_2"><span class="lab"><font color="Green" title="E70">A67</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">F</font> = <font color="Maroon" title="c21">F</font> as   <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c18">I</font> ;<br/><a NAME="E43:42_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A74</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds <br/> ( <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></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="42_1_1_1_2_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c23">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</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="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )</span><br/>

<a NAME="E1:42_1_1_1_2_7"/><span class="lab"><font color="Green" title="E72">A75</font></span>: 
<font color="Maroon" title="c23">x</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> <font color="Maroon" title="c22">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:42_1_1_1_2"><span class="lab"><font color="Green" title="E70">A67</font></span></a></span>;<br/>
<a NAME="E2:42_1_1_1_2_7"/><span class="lab"><font color="Green" title="E73">A76</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_7"><span class="lab"><font color="Green" title="E72">A75</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/>
<span class="lab"><font color="Green" title="E74">A77</font></span>: <font color="Maroon" title="c14">I0</font> = 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_7_2"><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="E1:42_1_1_1_2_7_2"/>
( <font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> or <font color="Maroon" title="c23">x</font> <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="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_7_2_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_7_2_1"/><span class="lab"><font color="Green" title="E75">A78</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )</span><br/><div class="add"><a NAME="E2:42_1_1_1_2_7_2_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c23">x</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="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_7"><span class="lab"><font color="Green" title="E74">A77</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:42_1_1_1_2_7_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A79</font></span>: 
<font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E2:42_1_1_1_2_7"><span class="lab"><font color="Green" title="E73">A76</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c24">p</font> = <font color="Maroon" title="c23">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E75">A78</font></span></a></span>;<br/><a NAME="E5:42_1_1_1_2_7_2_1"/><span class="lab"><font color="Green" title="E77">A80</font></span>: 
( <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c24">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c12">H</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c24">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c12">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>;<br/><a NAME="E6:42_1_1_1_2_7_2_1"/><span class="lab"><font color="Green" title="E78">A81</font></span>: 
( <font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c23">x</font> <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="c5">q</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="#E16:42_1_1_1_2"><span class="lab"><font color="Green" title="E56">A41</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E31:42_1_1_1_2"><span class="lab"><font color="Green" title="E65">A61</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E75">A78</font></span></a></span>;<br/><a NAME="E7:42_1_1_1_2_7_2_1"/>
<font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E78">A81</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">x0</font> = <font color="Maroon" title="c23">x</font> as   <a href="int_2.html#NM1" title="INT_2:NM.1">Prime</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T34" onmouseover="rs('nat_3/T34')" onmouseout="rh()">NAT_3:34</a></span>;<br/><a NAME="E9:42_1_1_1_2_7_2_1"/><span class="lab"><font color="Green" title="E79">A82</font></span>: 
<font color="Maroon" title="c9">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x0</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="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x0</font></span>)</span> <a href="xreal_0.html#K1" title="XREAL_0:func.1">-'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x0</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_poly.html#D6" onmouseover="rs('pre_poly/D6')" onmouseout="rh()">PRE_POLY:def 6</a></span>;<br/><a NAME="E10:42_1_1_1_2_7_2_1"/>
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="#E6:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E78">A81</font></span></a>, <a class="txt" href="#E14:42_1_1_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="pre_poly.html#D7" onmouseover="rs('pre_poly/D7')" onmouseout="rh()">PRE_POLY:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E11:42_1_1_1_2_7_2_1"/>
( <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E76">A79</font></span></a>, <a class="txt" href="#E5:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E77">A80</font></span></a>, <a class="ref" href="group_2.html#T56" onmouseover="rs('group_2/T56')" onmouseout="rh()">GROUP_2:56</a>, <a class="txt" href="#E37:42_1_1_1_2"><span class="lab"><font color="Green" title="E68">A64</font></span></a>, <a class="txt" href="#E9:42_1_1_1_2_7_2_1"><span class="lab"><font color="Green" title="E79">A82</font></span></a>, <a class="ref" href="nat_d.html#T40" onmouseover="rs('nat_d/T40')" onmouseout="rh()">NAT_D:40</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="42_1_1_1_2_7_2_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_7_2_2"/><span class="lab"><font color="Green" title="E75">A83</font></span>: 
<font color="Maroon" title="c23">x</font> <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="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.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"> ( <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )</span><br/><div class="add"><a NAME="E2:42_1_1_1_2_7_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A84</font></span>: 
<font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E2:42_1_1_1_2_7"><span class="lab"><font color="Green" title="E73">A76</font></span></a></span>;<br/><a NAME="E3:42_1_1_1_2_7_2_2"/>
<font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E75">A83</font></span></a>, <a class="txt" href="#E21:42_1_1_1_2"><span class="lab"><font color="Green" title="E60">A47</font></span></a></span>;<br/><a NAME="E4:42_1_1_1_2_7_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A85</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#NK13" title="NAT_3:NK.13">pfexp</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>;<br/><a NAME="E5:42_1_1_1_2_7_2_2"/><span class="lab"><font color="Green" title="E78">A86</font></span>: 
<font color="Maroon" title="c23">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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>, <a class="txt" href="#E1:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E75">A83</font></span></a></span>;<br/> <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> = 
 <a href="nat_3.html#K8" title="NAT_3:func.8">Product</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <font color="Maroon" title="c6">Gq</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E75">A83</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a>, <a class="txt" href="#E2:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E76">A84</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">Gq</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_3.html#T61" onmouseover="rs('nat_3/T61')" onmouseout="rh()">NAT_3:61</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E78">A86</font></span></a>, <a class="txt" href="#E4:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E77">A85</font></span></a>, <a class="ref" href="nat_3.html#D9" onmouseover="rs('nat_3/D9')" onmouseout="rh()">NAT_3:def 9</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:42_1_1_1_2_7_2_2"/>
( <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c23">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E76">A84</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_7_2_2"><span class="lab"><font color="Green" title="E75">A83</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</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><a NAME="E44:42_1_1_1_2"/><span class="lab"><font color="Green" title="E72">A87</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:42_1_1_1_2"><span class="lab"><font color="Green" title="E71">A74</font></span></a></span>;<br/><a NAME="E45:42_1_1_1_2"/><span class="lab"><font color="Green" title="E73">A88</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:42_1_1_1_2"><span class="lab"><font color="Green" title="E71">A74</font></span></a></span>;<br/><a NAME="E46:42_1_1_1_2"/>
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Maroon" title="c22">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">i</font> is  <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:42_1_1_1_2"><span class="lab"><font color="Green" title="E71">A74</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">F</font> = <font color="Maroon" title="c22">F</font> as   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Maroon" title="c18">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:42_1_1_1_2"><span class="lab"><font color="Green" title="E72">A87</font></span></a>, <a class="ref" href="group_7.html#D6" onmouseover="rs('group_7/D6')" onmouseout="rh()">GROUP_7:def 6</a>, <a class="txt" href="#E45:42_1_1_1_2"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="ref" href="group_7.html#D7" onmouseover="rs('group_7/D7')" onmouseout="rh()">GROUP_7:def 7</a>, <a class="ref" href="group_7.html#D8" onmouseover="rs('group_7/D8')" onmouseout="rh()">GROUP_7:def 8</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c24">FHKG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c12">H</font>,<font color="Maroon" title="c13">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c2">G</font><span class="kw"> such that </span><br/><a NAME="E49:42_1_1_1_2"/><span class="lab"><font color="Green" title="E74">A89</font></span>: 
( <font color="Maroon" title="c24">FHKG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  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 <font color="Maroon" title="c2">G</font>  st <font color="Olive" title="b1">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">H</font> &amp; <font color="Olive" title="b2">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">K</font> holds <br/><font color="Maroon" title="c24">FHKG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a></span>;<br/><span class="lab"><font color="Green" title="E75">A90</font></span>: <font color="Maroon" title="c14">I0</font> = 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c18">I</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E23:42_1_1_1_2"><span class="lab"><font color="Green" title="E62">A49</font></span></a>, <a class="txt" href="#E20:42_1_1_1_2"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="int_7.html#T15" onmouseover="rs('int_7/T15')" onmouseout="rh()">INT_7:15</a>, <a class="txt" href="#E36:42_1_1_1_2"><span class="lab"><font color="Green" title="E67">A63</font></span></a></span>
;<br/><a NAME="E51:42_1_1_1_2"/><span class="lab"><font color="Green" title="E76">A91</font></span>: 
 not <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_9"/>
<font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_2_9"/><span class="kw">then </span>
( <font color="Maroon" title="c5">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">I</font> &amp;  not <font color="Maroon" title="c5">q</font> <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="c5">q</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="#E50:42_1_1_1_2"><span class="lab"><font color="Green" title="E75">A90</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:42_1_1_1_2_9"/>
contradiction
 <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>; <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">consider </span><font color="Maroon" title="c25">FHK</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c12">H</font>,<font color="Maroon" title="c13">K</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E53:42_1_1_1_2"/><span class="lab"><font color="Green" title="E77">A92</font></span>: 
( <font color="Maroon" title="c25">FHK</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b1">x0</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><br/>  for <font color="Olive" title="b2">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">K</font><br/>  for <font color="Olive" title="b3">h</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">H</font>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">HFG0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x0</font> &amp; <font color="Olive" title="b1">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c15">F0</font> holds <br/><font color="Maroon" title="c25">FHK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Olive" title="b3">h</font>,<font color="Olive" title="b2">k</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T28" target="_self" onmouseover="rs('group_17/T28')" onmouseout="rh()">Th28</a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E38:42_1_1_1_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">HFG</font> = <font color="Maroon" title="c24">FHKG</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c25">FHK</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> ;<br/><a NAME="E55:42_1_1_1_2"/><span class="lab"><font color="Green" title="E78">A93</font></span>: 
<font color="Maroon" title="c26">HFG</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T27" onmouseover="rs('funct_2/T27')" onmouseout="rh()">FUNCT_2:27</a>, <a class="txt" href="#E49:42_1_1_1_2"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E53:42_1_1_1_2"><span class="lab"><font color="Green" title="E77">A92</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c27">HFG</font> = <font color="Maroon" title="c26">HFG</font> as   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> ;<br/><a NAME="E57:42_1_1_1_2"/><span class="lab"><font color="Green" title="E79">A94</font></span>: 
 for <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c18">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Olive" title="b1">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> ) holds <br/>( <font color="Olive" title="b1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font> &amp; <font color="Maroon" title="c27">HFG</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>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</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="42_1_1_1_2_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be  <font color="Maroon" title="c18">I</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</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">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Maroon" title="c28">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) implies ( <font color="Maroon" title="c28">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font> &amp; <font color="Maroon" title="c27">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c28">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_10"/><span class="lab"><font color="Green" title="E80">A95</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font> holds  <font color="Maroon" title="c28">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</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="c28">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font> &amp; <font color="Maroon" title="c27">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c28">x</font> )</span><br/>

<a NAME="E2:42_1_1_1_2_10"/><span class="kw">then </span>
<font color="Maroon" title="c28">x</font> <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 <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T29" target="_self" onmouseover="rs('group_17/T29')" onmouseout="rh()">Th29</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c29">x0</font> being  <font color="Maroon" title="c14">I0</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>, <font color="Maroon" title="c30">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c13">K</font><span class="kw"> such that </span><br/><a NAME="E4:42_1_1_1_2_10"/><span class="lab"><font color="Green" title="E81">A96</font></span>: 
( <font color="Maroon" title="c29">x0</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c15">F0</font> &amp; <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">x0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c30">k</font></span>)</span> &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> holds  <font color="Maroon" title="c29">x0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T30" target="_self" onmouseover="rs('group_17/T30')" onmouseout="rh()">Th30</a>, <a class="txt" href="#E51:42_1_1_1_2"><span class="lab"><font color="Green" title="E76">A91</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E38:42_1_1_1_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c31">h</font> = <font color="Maroon" title="c16">HFG0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">x0</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c12">H</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>, <a class="txt" href="#E4:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E81">A96</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c32">hh</font> = <font color="Maroon" title="c31">h</font>, <font color="Maroon" title="c33">kk</font> = <font color="Maroon" title="c30">k</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T42" onmouseover="rs('group_2/T42')" onmouseout="rh()">GROUP_2:42</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_10_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="c29">x0</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="c12">H</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c34">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="c34">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="c29">x0</font> implies <font color="Maroon" title="c34">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="c12">H</font> )</span><br/><span class="kw">assume </span><a NAME="E1:42_1_1_1_2_10_1"/>
<font color="Maroon" title="c34">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="c29">x0</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="c34">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="c12">H</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c35">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:42_1_1_1_2_10_1"/><span class="lab"><font color="Green" title="E82">A97</font></span>: 
( <font color="Maroon" title="c35">z</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="c29">x0</font> &amp; <font color="Maroon" title="c34">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">x0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c35">z</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/><span class="kw">reconsider </span><font color="Maroon" title="c36">z</font> = <font color="Maroon" title="c35">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_10_1"><span class="lab"><font color="Green" title="E82">A97</font></span></a></span>;<br/><a NAME="E5:42_1_1_1_2_10_1"/><span class="lab"><font color="Green" title="E83">A98</font></span>: 
<font color="Maroon" title="c29">x0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c36">z</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c36">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E81">A96</font></span></a></span>;<br/><a NAME="E6:42_1_1_1_2_10_1"/>
<font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c36">z</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c12">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_1_1_1_2_10_1"/>
<font color="Maroon" title="c34">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="c12">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_10_1"><span class="lab"><font color="Green" title="E82">A97</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a>, <a class="txt" href="#E5:42_1_1_1_2_10_1"><span class="lab"><font color="Green" title="E83">A98</font></span></a>, <a class="ref" href="group_2.html#T40" onmouseover="rs('group_2/T40')" onmouseout="rh()">GROUP_2:40</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="c34">x0</font> = <font color="Maroon" title="c29">x0</font> as  <font color="Maroon" title="c14">I0</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E9:42_1_1_1_2_10"/><span class="lab"><font color="Green" title="E82">A99</font></span>: 
<font color="Maroon" title="c16">HFG0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c34">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c34">x0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E4:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E81">A96</font></span></a></span>;<br/>
<a NAME="E10:42_1_1_1_2_10"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">H</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">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/>
<a NAME="E11:42_1_1_1_2_10"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c34">x0</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">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c35">xx0</font> = <font color="Maroon" title="c34">x0</font> as  <font color="Maroon" title="c14">I0</font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E13:42_1_1_1_2_10"/><span class="lab"><font color="Green" title="E83">A100</font></span>: 
 <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c34">x0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c35">xx0</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T32" target="_self" onmouseover="rs('group_17/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<span class="kw">thus </span><a NAME="E14:42_1_1_1_2_10"/>
<font color="Maroon" title="c28">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c23">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T29" target="_self" onmouseover="rs('group_17/T29')" onmouseout="rh()">Th29</a>, <a class="txt" href="#E1:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E80">A95</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"> <font color="Maroon" title="c27">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c28">x</font></span><br/>

<a NAME="E15:42_1_1_1_2_10"/><span class="lab"><font color="Green" title="E84">A101</font></span>: 
( <font color="Maroon" title="c32">hh</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c12">H</font> &amp; <font color="Maroon" title="c33">kk</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c13">K</font> )
 
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c27">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> = 
<font color="Maroon" title="c24">FHKG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c25">FHK</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" onmouseover="rs('funct_2/T15')" onmouseout="rh()">FUNCT_2:15</a>, <a class="ref" href="group_17.html#T29" target="_self" onmouseover="rs('group_17/T29')" onmouseout="rh()">Th29</a>, <a class="txt" href="#E1:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E80">A95</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">FHKG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c32">hh</font>,<font color="Maroon" title="c33">kk</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:42_1_1_1_2"><span class="lab"><font color="Green" title="E77">A92</font></span></a>, <a class="txt" href="#E4:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E81">A96</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c32">hh</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c33">kk</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:42_1_1_1_2"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E15:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E84">A101</font></span></a></span>
<br/>.= 
 <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c28">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E82">A99</font></span></a>, <a class="txt" href="#E13:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E83">A100</font></span></a>, <a class="ref" href="group_17.html#T33" target="_self" onmouseover="rs('group_17/T33')" onmouseout="rh()">Th33</a>, <a class="txt" href="#E51:42_1_1_1_2"><span class="lab"><font color="Green" title="E76">A91</font></span></a>, <a class="txt" href="#E6:42_1_1_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="txt" href="#E38:42_1_1_1_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a>, <a class="txt" href="#E4:42_1_1_1_2_10"><span class="lab"><font color="Green" title="E81">A96</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="E58:42_1_1_1_2"/>
 for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">t</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">s</font>, <font color="Maroon" title="c29">t</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c18">I</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="c28">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">t</font> implies  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_11"/><span class="lab"><font color="Green" title="E80">A102</font></span>: 
<font color="Maroon" title="c28">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c29">t</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E2:42_1_1_1_2_11"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c23">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E3:42_1_1_1_2_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A103</font></span>: 
( <font color="Maroon" title="c28">s</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> <font color="Maroon" title="c23">F</font> &amp; <font color="Maroon" title="c29">t</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> <font color="Maroon" title="c23">F</font> )
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_11_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="E1:42_1_1_1_2_11_1"/>
( ( <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> &amp; <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> ) or  not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> or  not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_11_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_11_1_1"/>
( <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> &amp; <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">ss</font> = <font color="Maroon" title="c28">s</font>, <font color="Maroon" title="c31">tt</font> = <font color="Maroon" title="c29">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> ;<br/><a NAME="E3:42_1_1_1_2_11_1_1"/><span class="lab"><font color="Green" title="E82">A104</font></span>: 
<font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><a NAME="E4:42_1_1_1_2_11_1_1"/><span class="lab"><font color="Green" title="E83">A105</font></span>: 
<font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E84">A106</font></span>: <font color="Maroon" title="c14">I0</font> = 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a></span>
;<br/><a NAME="E6:42_1_1_1_2_11_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c30">ss</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="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E7:42_1_1_1_2_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A107</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">ss</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E3:42_1_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E82">A104</font></span></a></span>;<br/><a NAME="E8:42_1_1_1_2_11_1_1"/>
 not <font color="Maroon" title="c31">tt</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="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E84">A106</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E9:42_1_1_1_2_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A108</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c31">tt</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E4:42_1_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E83">A105</font></span></a></span>;<br/><a NAME="E10:42_1_1_1_2_11_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">ss</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c31">tt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c12">H</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a>, <a class="txt" href="#E1:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E80">A102</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:42_1_1_1_2_11_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42_1_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E85">A107</font></span></a>, <a class="txt" href="#E9:42_1_1_1_2_11_1_1"><span class="lab"><font color="Green" title="E86">A108</font></span></a>, <a class="ref" href="group_2.html#T44" onmouseover="rs('group_2/T44')" onmouseout="rh()">GROUP_2:44</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="42_1_1_1_2_11_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_11_1_2"/><span class="lab"><font color="Green" title="E82">A109</font></span>: 
(  not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> or  not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:42_1_1_1_2_11_1_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_11_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_11_1_2_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="E1:42_1_1_1_2_11_1_2_1_1"/>
(  not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> or  not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_11_1_2"><span class="lab"><font color="Green" title="E82">A109</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_2_11_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A110</font></span>: 
 not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E84">A111</font></span>: 
<font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><a NAME="E3:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E85">A112</font></span>: 
<font color="Maroon" title="c28">s</font> <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="c5">q</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="#E1:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A110</font></span></a>, <a class="txt" href="#E50:42_1_1_1_2"><span class="lab"><font color="Green" title="E75">A90</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E2:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E84">A111</font></span></a></span>;<br/><a NAME="E5:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A113</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A112</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><a NAME="E6:42_1_1_1_2_11_1_2_1_1_1"/>
<font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_11_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_11_1_2_1_1_1_1"/>
 not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_2_11_1_2_1_1_1_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">I</font> or <font color="Maroon" title="c29">t</font> <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="c5">q</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E50:42_1_1_1_2"><span class="lab"><font color="Green" title="E75">A90</font></span></a></span>;<br/>
<a NAME="E3:42_1_1_1_2_11_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c29">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">hence </span><a NAME="E4:42_1_1_1_2_11_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E80">A102</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E3:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E85">A112</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">tt</font> = <font color="Maroon" title="c29">t</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> ;<br/><a NAME="E8:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A114</font></span>: 
<font color="Maroon" title="c30">tt</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><font color="Maroon" title="c14">I0</font> = 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a>, <a class="txt" href="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a></span>
;<br/><a NAME="E10:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c30">tt</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="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E11:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A115</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">tt</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E8:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A114</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">S1</font> = <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">tt</font> as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c12">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>;<br/><a NAME="E13:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E89">A116</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/><a NAME="E14:42_1_1_1_2_11_1_2_1_1_1"/><span class="lab"><font color="Green" title="E90">A117</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><a NAME="E15:42_1_1_1_2_11_1_2_1_1_1"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c12">H</font> <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="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><a NAME="E16:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T44" onmouseover="rs('group_2/T44')" onmouseout="rh()">GROUP_2:44</a></span>;<br/><a NAME="E17:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a>, <a class="txt" href="#E14:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E90">A117</font></span></a></span>;<br/><a NAME="E18:42_1_1_1_2_11_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E19:42_1_1_1_2_11_1_2_1_1_1"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A113</font></span></a>, <a class="txt" href="#E11:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A115</font></span></a>, <a class="txt" href="#E13:42_1_1_1_2_11_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A116</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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="42_1_1_1_2_11_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A118</font></span>: 
 not <font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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">  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><div class="add"><a NAME="E2:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A119</font></span>: 
<font color="Maroon" title="c29">t</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><a NAME="E3:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E85">A120</font></span>: 
<font color="Maroon" title="c29">t</font> <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="c5">q</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="#E1:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A118</font></span></a>, <a class="txt" href="#E50:42_1_1_1_2"><span class="lab"><font color="Green" title="E75">A90</font></span></a>, <a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E4:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E2:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A119</font></span></a></span>;<br/><a NAME="E5:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A121</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A120</font></span></a>, <a class="ref" href="funcop_1.html#T7" onmouseover="rs('funcop_1/T7')" onmouseout="rh()">FUNCOP_1:7</a></span>;<br/><a NAME="E6:42_1_1_1_2_11_1_2_1_1_2"/>
<font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_1_1_1_2_11_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_1_1_1_2_11_1_2_1_1_2_1"/>
 not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c14">I0</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"> contradiction</span><br/>

<a NAME="E2:42_1_1_1_2_11_1_2_1_1_2_1"/><span class="kw">then </span>
(  not <font color="Maroon" title="c28">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c18">I</font> or <font color="Maroon" title="c28">s</font> <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="c5">q</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="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a>, <a class="txt" href="#E50:42_1_1_1_2"><span class="lab"><font color="Green" title="E75">A90</font></span></a></span>;<br/>
<a NAME="E3:42_1_1_1_2_11_1_2_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c28">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</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">hence </span><a NAME="E4:42_1_1_1_2_11_1_2_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E80">A102</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E3:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A120</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">ss</font> = <font color="Maroon" title="c28">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c14">I0</font> ;<br/><a NAME="E8:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E87">A122</font></span>: 
<font color="Maroon" title="c30">ss</font> <a href="tarski.html#R2" title="TARSKI: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="c15">F0</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42_1_1_1_2_11"><span class="lab"><font color="Green" title="E81">A103</font></span></a>, <a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>;<br/><font color="Maroon" title="c14">I0</font> = 
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <font color="Maroon" title="c9">h</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="int_7.html#T16" onmouseover="rs('int_7/T16')" onmouseout="rh()">INT_7:16</a>, <a class="ref" href="int_7.html#D1" onmouseover="rs('int_7/D1')" onmouseout="rh()">INT_7:def 1</a>, <a class="txt" href="#E19:42_1_1_1_2"><span class="lab"><font color="Green" title="E58">A43</font></span></a>, <a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</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="#E15:42_1_1_1_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI:2</a></span>
;<br/><a NAME="E10:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c30">ss</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="c5">q</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a> <font color="Maroon" title="c13">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" onmouseover="rs('xboole_0/D5')" onmouseout="rh()">XBOOLE_0:def 5</a></span>;<br/><a NAME="E11:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A123</font></span>: 
<font color="Maroon" title="c23">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">ss</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a>, <a class="txt" href="#E8:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A122</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c31">S1</font> = <font color="Maroon" title="c15">F0</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c30">ss</font> as   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c12">H</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:42_1_1_1_2"><span class="lab"><font color="Green" title="E66">A62</font></span></a></span>;<br/><a NAME="E13:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A124</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:42_1_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="xboole_1.html#T26" onmouseover="rs('xboole_1/T26')" onmouseout="rh()">XBOOLE_1:26</a>, <a class="ref" href="group_2.html#D5" onmouseover="rs('group_2/D5')" onmouseout="rh()">GROUP_2:def 5</a></span>;<br/><a NAME="E14:42_1_1_1_2_11_1_2_1_1_2"/><span class="lab"><font color="Green" title="E90">A125</font></span>: 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c13">K</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><a NAME="E15:42_1_1_1_2_11_1_2_1_1_2"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c12">H</font> <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="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T46" onmouseover="rs('group_2/T46')" onmouseout="rh()">GROUP_2:46</a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><a NAME="E16:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T44" onmouseover="rs('group_2/T44')" onmouseout="rh()">GROUP_2:44</a></span>;<br/><a NAME="E17:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font> <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="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a>, <a class="txt" href="#E14:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A125</font></span></a></span>;<br/><a NAME="E18:42_1_1_1_2_11_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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="c13">K</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c31">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T31" onmouseover="rs('zfmisc_1/T31')" onmouseout="rh()">ZFMISC_1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E19:42_1_1_1_2_11_1_2_1_1_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c28">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c23">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c29">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E86">A121</font></span></a>, <a class="txt" href="#E11:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A123</font></span></a>, <a class="txt" href="#E13:42_1_1_1_2_11_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A124</font></span></a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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><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="E59:42_1_1_1_2"/>
 ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c2">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c2">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c2">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:42_1_1_1_2"><span class="lab"><font color="Green" title="E69">A65</font></span></a>, <a class="txt" href="#E43:42_1_1_1_2"><span class="lab"><font color="Green" title="E71">A74</font></span></a>, <a class="txt" href="#E55:42_1_1_1_2"><span class="lab"><font color="Green" title="E78">A93</font></span></a>, <a class="txt" href="#E49:42_1_1_1_2"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E53:42_1_1_1_2"><span class="lab"><font color="Green" title="E77">A92</font></span></a>, <a class="txt" href="#E57:42_1_1_1_2"><span class="lab"><font color="Green" title="E79">A94</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><span class="kw">end;</span></div>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E45">A126</font></span>: 
 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>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E2:42"><span class="lab"><font color="Green" title="E44">A2</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a>  <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</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"> (  <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies  ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E4:42"/><span class="lab"><font color="Green" title="E46">A127</font></span>: 
 <a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )</span><br/>

<a NAME="E5:42"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:42_2"/>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p2">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p3">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span></span>)</span> <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"> contradiction</span><br/>

<a NAME="E2:42_2"/><span class="kw">then </span>
 <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:42_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E46">A127</font></span></a>, <a class="ref" href="nat_3.html#T57" onmouseover="rs('nat_3/T57')" onmouseout="rh()">NAT_3:57</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="E6:42"/>
 ex <font color="Olive" title="b1">I</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  ex <font color="Olive" title="b2">F</font> being   <a href="group_7.html#V2" title="GROUP_7:attr.2">Group-like</a>   <a href="group_7.html#V3" title="GROUP_7:attr.3">associative</a>   <a href="group_7.html#V4" title="GROUP_7:attr.4">commutative</a>  <a href="group_7.html#NM1" title="GROUP_7:NM.1">multMagma-Family</a> of <font color="Olive" title="b1">I</font> ex <font color="Olive" title="b3">HFG</font> being   <a href="group_6.html#NM2" title="GROUP_6:NM.2">Homomorphism</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_poly.html#K22" title="PRE_POLY:func.22">support</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b4">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font> is   <a href="algstr_0.html#V15" title="ALGSTR_0:attr.15">strict</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font> &amp;  <a href="struct_0.html#K7" title="STRUCT_0:func.7">card</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="nat_3.html#K13" title="NAT_3:func.13">prime_factorization</a> <span class="p2">(<span class="default"><a href="group_1.html#K7" title="GROUP_1:func.7">card</a> <font color="Maroon" title="c1">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">p</font> ) ) &amp; (  for <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font>  st <font color="Olive" title="b4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b5">q</font> holds <br/> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b4">p</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Olive" title="b3">HFG</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">x</font> being  <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#V4" title="RELAT_1:attr.4">-defined</a>   the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="relat_1.html#V5" title="RELAT_1:attr.5">-valued</a>   <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st (  for <font color="Olive" title="b5">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">I</font> holds  <font color="Olive" title="b4">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b5">p</font> ) holds <br/>( <font color="Olive" title="b4">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a>  <a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b3">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b4">x</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E45">A126</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/>


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