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<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> 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="b3">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="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="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="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="c1">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="Olive" title="b1">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="b2">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="b2">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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; (  for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <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 <br/>( (  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> ) &amp; <font color="Olive" title="b3">y</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> ) ) &amp; (  for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</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="b3">x1</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> ) &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="b1">I</font> holds  <font color="Olive" title="b4">x2</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> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x1</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">x2</font> holds <br/><font color="Olive" title="b3">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">x2</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:43"/>
 <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> 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="b3">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="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="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="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="c1">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="Olive" title="b1">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="b2">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="b2">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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; (  for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <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 <br/>( (  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> ) &amp; <font color="Olive" title="b3">y</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> ) ) &amp; (  for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</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="b3">x1</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> ) &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="b1">I</font> holds  <font color="Olive" title="b4">x2</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> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x1</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">x2</font> holds <br/><font color="Olive" title="b3">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">x2</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c2">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> , <font color="Maroon" title="c3">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="c2">I</font>, <font color="Maroon" title="c4">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="c3">F</font></span>)</span>,<font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E43">A1</font></span>: 
( <font color="Maroon" title="c2">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="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds <br/> ( <font color="Maroon" title="c3">F</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="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="Maroon" title="c3">F</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="c1">G</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="c2">I</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="c3">F</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="c3">F</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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; <font color="Maroon" title="c4">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  <font color="Maroon" title="c2">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="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="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c3">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="c3">F</font> &amp; <font color="Maroon" title="c4">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#T34" target="_self" onmouseover="rs('group_17/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E4:43"/><span class="lab"><font color="Green" title="E44">A2</font></span>: 
 for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <font color="Olive" title="b2">x</font> being  <font color="Maroon" title="c2">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="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 <br/>( (  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="c2">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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> ) &amp; <font color="Olive" title="b1">y</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> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being  <font color="Maroon" title="c2">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="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 <br/>( (  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="c2">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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b2">p</font> ) &amp; <font color="Maroon" title="c5">y</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><br/>

<a NAME="E1:43_1"/>
<font color="Maroon" title="c5">y</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="c1">G</font>
 
;<br/>
<a NAME="E2:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">HFG</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E45">A3</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <font color="Maroon" title="c3">F</font></span>)</span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T11" onmouseover="rs('funct_2/T11')" onmouseout="rh()">FUNCT_2:11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c6">x</font> as  <font color="Maroon" title="c2">I</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="group_17.html#E29"><span class="lab"><font color="Green" title="E29">Lm6</font></span></a></span>;<br/>
<a NAME="E6:43_1"/><span class="lab"><font color="Green" title="E46">A4</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="c2">I</font> holds  <font color="Maroon" title="c7">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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">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="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c9">R</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> <span class="kw"> such that </span><br/><a NAME="E2:43_1_1"/><span class="lab"><font color="Green" title="E47">A5</font></span>: 
( <font color="Maroon" title="c9">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</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="c9">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="group_17.html#E30"><span class="lab"><font color="Green" title="E30">Lm7</font></span></a>, <a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:43_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c8">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E47">A5</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="E7:43_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c7">x</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="c1">G</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c8">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="c7">x</font> or <font color="Maroon" title="c8">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="c1">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_1_2"/>
<font color="Maroon" title="c8">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="c7">x</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="c8">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="c1">G</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:43_1_2"/><span class="lab"><font color="Green" title="E47">A6</font></span>: 
( <font color="Maroon" title="c9">i</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="c7">x</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</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="c10">i</font> = <font color="Maroon" title="c9">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E47">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">R</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> <span class="kw"> such that </span><br/><a NAME="E6:43_1_2"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
( <font color="Maroon" title="c11">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c10">i</font> &amp; <font color="Maroon" title="c7">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">i</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="c11">R</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="group_17.html#E30"><span class="lab"><font color="Green" title="E30">Lm7</font></span></a></span>;<br/>
<a NAME="E7:43_1_2"/>
<font color="Maroon" title="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c10">i</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/>
<a NAME="E8:43_1_2"/><span class="kw">then </span>
 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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Maroon" title="c10">i</font></span>)</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="c1">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/>
<span class="kw">hence </span><a NAME="E9:43_1_2"/>
<font color="Maroon" title="c8">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="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E47">A6</font></span></a>, <a class="txt" href="#E6:43_1_2"><span class="lab"><font color="Green" title="E48">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">x</font> = <font color="Maroon" title="c7">x</font> as  <font color="Maroon" title="c2">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="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> <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/>
<span class="kw">take </span>
<font color="Maroon" title="c8">x</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <font color="Maroon" title="c8">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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &amp; <font color="Maroon" title="c5">y</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="c8">x</font> )</span><br/>

<span class="kw">thus </span><a NAME="E9:43_1"/>
( (  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="c2">I</font> holds  <font color="Maroon" title="c8">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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &amp; <font color="Maroon" title="c5">y</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="c8">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E45">A3</font></span></a>, <a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E46">A4</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="43_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being  <font color="Maroon" title="c2">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="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="b3">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">I</font> holds  <font color="Olive" title="b1">x1</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> ) &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="c2">I</font> holds  <font color="Olive" title="b2">x2</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b3">p</font> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b1">x1</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">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b2">x2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> be  <font color="Maroon" title="c2">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="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>; <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="c2">I</font> holds  <font color="Maroon" title="c5">x1</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &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="c2">I</font> holds  <font color="Maroon" title="c6">x2</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x1</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="c6">x2</font> implies <font color="Maroon" title="c5">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E45">A8</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="c2">I</font> holds  <font color="Maroon" title="c5">x1</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &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="c2">I</font> holds  <font color="Maroon" title="c6">x2</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="c3">F</font> <a href="group_7.html#K1" title="GROUP_7:func.1">.</a> <font color="Olive" title="b1">p</font> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Maroon" title="c5">x1</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="c6">x2</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="c5">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">x2</font></span><br/><a NAME="E2:43_2"/>
( <font color="Maroon" title="c5">x1</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="c3">F</font> &amp; <font color="Maroon" title="c4">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</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="c5">x1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/><a NAME="E3:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<font color="Maroon" title="c4">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">HFG</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a></span>;<br/><a NAME="E4:43_2"/>
( <font color="Maroon" title="c5">x1</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="c3">F</font></span>)</span> &amp; <font color="Maroon" title="c6">x2</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="c3">F</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="struct_0.html#D5" onmouseover="rs('struct_0/D5')" onmouseout="rh()">STRUCT_0:def 5</a></span>;<br/><span class="kw">hence </span><a NAME="E5:43_2"/>
<font color="Maroon" title="c5">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c6">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="ref" href="funct_2.html#T19" onmouseover="rs('funct_2/T19')" onmouseout="rh()">FUNCT_2:19</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="E6:43"/>
 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> 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="b3">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="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="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="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="c1">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="Olive" title="b1">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="b2">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="b2">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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; (  for <font color="Olive" title="b3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  ex <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 <br/>( (  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> ) &amp; <font color="Olive" title="b3">y</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> ) ) &amp; (  for <font color="Olive" title="b3">x1</font>, <font color="Olive" title="b4">x2</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="b3">x1</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> ) &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="b1">I</font> holds  <font color="Olive" title="b4">x2</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> ) &amp;  <a href="group_17.html#K1" title="GROUP_17:func.1">Product</a> <font color="Olive" title="b3">x1</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">x2</font> holds <br/><font color="Olive" title="b3">x1</font> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Olive" title="b4">x2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E43">A1</font></span></a>, <a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E44">A2</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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