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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <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">  for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</a> of <font color="Maroon" title="c1">G</font>  st (  for <font color="Olive" title="b3">x</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">a</font>, <font color="Olive" title="b5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b5">b</font> ) ) &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">A</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="Olive" title="b2">B</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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/> ex <font color="Olive" title="b3">h</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="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b3">h</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  &amp; (  for <font color="Olive" title="b4">a</font>, <font color="Olive" title="b5">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>  st <font color="Olive" title="b4">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b5">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b3">h</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="b4">a</font>,<font color="Olive" title="b5">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="b4">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b5">b</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="group_3.html#V1" title="GROUP_3:attr.1">normal</a>   <a href="group_2.html#M1" title="GROUP_2:mode.1">Subgroup</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"> ( (  for <font color="Olive" title="b1">x</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">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="c1">G</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</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="c3">B</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> ) ) &amp;  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</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="c3">B</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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies  ex <font color="Olive" title="b1">h</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">h</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="c1">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="c2">A</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="c3">B</font> holds <br/><font color="Olive" title="b1">h</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:17"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 for <font color="Olive" title="b1">x</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">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="c1">G</font> st <br/>( <font color="Olive" title="b2">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</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="c3">B</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</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="c3">B</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="c1">G</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or  ex <font color="Olive" title="b1">h</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">h</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="c1">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="c2">A</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="c3">B</font> holds <br/><font color="Olive" title="b1">h</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:17"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">A</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="c3">B</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="c1">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">  ex <font color="Olive" title="b1">h</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">h</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="c1">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="c2">A</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="c3">B</font> holds <br/><font color="Olive" title="b1">h</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b2">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; $1 <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="b1">x</font>,<font color="Olive" title="b2">y</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font> );<br/>
<a NAME="E3:17"/><span class="lab"><font color="Green" title="E20">A3</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>  ex <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Olive" title="b2">w</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></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">  ex <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">w</font>]</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E2:17_1"/><span class="lab"><font color="Green" title="E21">A4</font></span>: 
<font color="Maroon" title="c4">z</font> <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="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</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="topalg_4.html#T1" onmouseover="rs('topalg_4/T1')" onmouseout="rh()">TOPALG_4:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T41" onmouseover="rs('group_2/T41')" onmouseout="rh()">GROUP_2:41</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="E4:17_1"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y1</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>
 
;<br/>
<a NAME="E5:17_1"/>
( <font color="Maroon" title="c7">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E6:17_1"/>
 ex <font color="Olive" title="b1">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">z</font>,<font color="Olive" title="b1">w</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_1"><span class="lab"><font color="Green" title="E21">A4</font></span></a>, <a class="txt" href="#E4:17_1"><span class="lab"><font color="Green" title="E22">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>
<span class="kw">consider </span><font color="Maroon" title="c4">h</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:17"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">z</font>,<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S3" onmouseover="rs('funct_2/S3')" onmouseout="rh()">FUNCT_2:sch 3</a>(<a class="txt" href="#E3:17"><span class="lab"><font color="Green" title="E20">A3</font></span></a>);<br/></span>
<a NAME="E6:17"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 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="c1">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="c2">A</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="c3">B</font> holds <br/><font color="Maroon" title="c4">h</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">b</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"> ( <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c4">h</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="c5">a</font>,<font color="Maroon" title="c6">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="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:17_2"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
( <font color="Maroon" title="c5">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c6">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</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="c4">h</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="c5">a</font>,<font color="Maroon" title="c6">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="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">a1</font> = <font color="Maroon" title="c5">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">b1</font> = <font color="Maroon" title="c6">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17_2"><span class="lab"><font color="Green" title="E23">A8</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:17_2"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
( <font color="Maroon" title="c9">x</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">y</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</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> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c10">y</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</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="Maroon" title="c9">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/>
<span class="lab"><font color="Green" title="E25">A10</font></span>: <font color="Maroon" title="c7">a1</font> = 
<span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1

<br/>.= 
<font color="Maroon" title="c9">x</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a>, <a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>
;<br/>
<font color="Maroon" title="c8">b1</font> = 
<span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c7">a1</font>,<font color="Maroon" title="c8">b1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2

<br/>.= 
<font color="Maroon" title="c10">y</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T44" onmouseover="rs('finseq_1/T44')" onmouseout="rh()">FINSEQ_1:44</a>, <a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E24">A9</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E8:17_2"/>
<font color="Maroon" title="c4">h</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="c5">a</font>,<font color="Maroon" title="c6">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="Maroon" title="c5">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_2"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="txt" href="#E6:17_2"><span class="lab"><font color="Green" title="E25">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_3"><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">z1</font>, <font color="Olive" title="b2">z2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">z1</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Olive" title="b2">z2</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">z2</font> holds <br/><font color="Olive" title="b1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">z2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">z1</font>, <font color="Maroon" title="c6">z2</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="c5">z1</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c6">z2</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z2</font> implies <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_3"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
( <font color="Maroon" title="c5">z1</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c6">z2</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> <span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z2</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">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">y1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E3:17_3"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c5">z1</font> <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="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">x2</font>, <font color="Maroon" title="c10">y2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E5:17_3"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
( <font color="Maroon" title="c9">x2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c10">y2</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c6">z2</font> <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="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> &amp; <font color="Maroon" title="c4">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E1:17_3"><span class="lab"><font color="Green" title="E23">A11</font></span></a></span>;<br/><a NAME="E6:17_3"/>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y2</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T14" onmouseover="rs('group_1/T14')" onmouseout="rh()">GROUP_1:14</a>, <a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E1:17_3"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E24">A12</font></span></a></span>;<br/><a NAME="E7:17_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>;<br/><a NAME="E8:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T13" onmouseover="rs('group_1/T13')" onmouseout="rh()">GROUP_1:13</a></span>;<br/><a NAME="E9:17_3"/>
<font color="Maroon" title="c9">x2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/><a NAME="E10:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">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 <font color="Maroon" title="c2">A</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a>, <a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E24">A12</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/><a NAME="E11:17_3"/>
<font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a>  <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="group_2.html#T51" onmouseover="rs('group_2/T51')" onmouseout="rh()">GROUP_2:51</a></span>;<br/><a NAME="E12:17_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">y2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</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="E13:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">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 <font color="Maroon" title="c2">A</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="c3">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:17_3"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E10:17_3"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E14:17_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x2</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E15:17_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T13" onmouseover="rs('group_1/T13')" onmouseout="rh()">GROUP_1:13</a></span>;<br/><a NAME="E16:17_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c7">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" onmouseover="rs('group_1/D4')" onmouseout="rh()">GROUP_1:def 4</a></span>;<br/><a NAME="E17:17_3"/>
<font color="Maroon" title="c10">y2</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y1</font> <a href="group_1.html#K2" title="GROUP_1:func.2">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a>, <a class="txt" href="#E8:17_3"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E13:17_3"><span class="lab"><font color="Green" title="E28">A16</font></span></a></span>;<br/><a NAME="E18:17_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">y1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#T14" onmouseover="rs('group_1/T14')" onmouseout="rh()">GROUP_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E19:17_3"/>
<font color="Maroon" title="c5">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:17_3"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E5:17_3"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="txt" href="#E16:17_3"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="group_1.html#D4" onmouseover="rs('group_1/D4')" onmouseout="rh()">GROUP_1: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>
<a NAME="E8:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A18</font></span>: 
<font color="Maroon" title="c4">h</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T19" onmouseover="rs('funct_2/T19')" onmouseout="rh()">FUNCT_2:19</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_4"><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">w</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">w</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> holds <br/><font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">w</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="c5">w</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> implies <font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font> )</span><br/><span class="kw">assume </span><a NAME="E1:17_4"/>
<font color="Maroon" title="c5">w</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>
 ; <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">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">g</font> = <font color="Maroon" title="c5">w</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font><span class="kw"> such that </span><br/><a NAME="E4:17_4"/><span class="lab"><font color="Green" title="E24">A19</font></span>: 
( <font color="Maroon" title="c7">a</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">b</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:17"><span class="lab"><font color="Green" title="E18">A1</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">a1</font> = <font color="Maroon" title="c7">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_4"><span class="lab"><font color="Green" title="E24">A19</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c10">b1</font> = <font color="Maroon" title="c8">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_4"><span class="lab"><font color="Green" title="E24">A19</font></span></a></span>;<br/><a NAME="E7:17_4"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</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="Maroon" title="c7">a</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E4:17_4"><span class="lab"><font color="Green" title="E24">A19</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:17_4"/>
<font color="Maroon" title="c5">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:17_4"><span class="lab"><font color="Green" title="E24">A19</font></span></a>, <a class="ref" href="funct_2.html#T112" onmouseover="rs('funct_2/T112')" onmouseout="rh()">FUNCT_2:112</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="E10:17"/><span class="kw">then </span>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E11:17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A20</font></span>: 
<font color="Maroon" title="c4">h</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#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<a NAME="E12:17"/>
 for <font color="Olive" title="b1">z</font>, <font color="Olive" title="b2">w</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span> holds  <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="17_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">z</font>, <font color="Maroon" title="c6">w</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></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="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c8">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E2:17_5"/><span class="lab"><font color="Green" title="E25">A21</font></span>: 
<font color="Maroon" title="c5">z</font> <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="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</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="topalg_4.html#T1" onmouseover="rs('topalg_4/T1')" onmouseout="rh()">TOPALG_4:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x1</font> = <font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c10">y1</font> = <font color="Maroon" title="c8">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T41" onmouseover="rs('group_2/T41')" onmouseout="rh()">GROUP_2:41</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">consider </span><font color="Maroon" title="c11">a</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c12">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">B</font><span class="kw"> such that </span><br/><a NAME="E5:17_5"/><span class="lab"><font color="Green" title="E26">A22</font></span>: 
<font color="Maroon" title="c6">w</font> <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="Maroon" title="c11">a</font>,<font color="Maroon" title="c12">b</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="topalg_4.html#T1" onmouseover="rs('topalg_4/T1')" onmouseout="rh()">TOPALG_4:1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">a1</font> = <font color="Maroon" title="c11">a</font>, <font color="Maroon" title="c14">b1</font> = <font color="Maroon" title="c12">b</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T41" onmouseover="rs('group_2/T41')" onmouseout="rh()">GROUP_2:41</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="E7:17_5"/><span class="lab"><font color="Green" title="E27">A23</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a></span>;<br/>
<span class="lab"><font color="Green" title="E28">A24</font></span>: <font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font> = 
<span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c11">a</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">y</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c12">b</font></span>)</span></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="#E2:17_5"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E5:17_5"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="ref" href="group_7.html#T29" onmouseover="rs('group_7/T29')" onmouseout="rh()">GROUP_7:29</a></span>
<br/>.= 
<span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span></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_2.html#T43" onmouseover="rs('group_2/T43')" onmouseout="rh()">GROUP_2:43</a>, <a class="txt" href="#E7:17_5"><span class="lab"><font color="Green" title="E27">A23</font></span></a></span>
;<br/>
<a NAME="E9:17_5"/><span class="lab"><font color="Green" title="E29">A25</font></span>: 
( <font color="Maroon" title="c9">x1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c13">a1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font> )
 
;<br/>
<a NAME="E10:17_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A26</font></span>: 
<font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<a NAME="E11:17_5"/><span class="lab"><font color="Green" title="E31">A27</font></span>: 
( <font color="Maroon" title="c10">y1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c14">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font> )
 
;<br/>
<a NAME="E12:17_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A28</font></span>: 
<font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font> <a href="struct_0.html#R1" title="STRUCT_0:pred.1">in</a> <font color="Maroon" title="c3">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_2.html#T50" onmouseover="rs('group_2/T50')" onmouseout="rh()">GROUP_2:50</a></span>;<br/>
<span class="lab"><font color="Green" title="E33">A29</font></span>: <font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E8:17_5"><span class="lab"><font color="Green" title="E28">A24</font></span></a>, <a class="txt" href="#E10:17_5"><span class="lab"><font color="Green" title="E30">A26</font></span></a>, <a class="txt" href="#E12:17_5"><span class="lab"><font color="Green" title="E32">A28</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c13">a1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_17.html#T11" target="_self" onmouseover="rs('group_17/T11')" onmouseout="rh()">Th11</a>, <a class="txt" href="#E2:17"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E9:17_5"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E11:17_5"><span class="lab"><font color="Green" title="E31">A27</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y1</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c14">b1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D3" onmouseover="rs('group_1/D3')" onmouseout="rh()">GROUP_1:def 3</a></span>
;<br/>
<a NAME="E14:17_5"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:17_5"><span class="lab"><font color="Green" title="E25">A21</font></span></a>, <a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E9:17_5"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E11:17_5"><span class="lab"><font color="Green" title="E31">A27</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:17_5"/>
<font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">z</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c6">w</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">z</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">h</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">w</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:17_5"><span class="lab"><font color="Green" title="E33">A29</font></span></a>, <a class="txt" href="#E5:17_5"><span class="lab"><font color="Green" title="E26">A22</font></span></a>, <a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E9:17_5"><span class="lab"><font color="Green" title="E29">A25</font></span></a>, <a class="txt" href="#E11:17_5"><span class="lab"><font color="Green" title="E31">A27</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E13:17"/><span class="kw">then </span>
<font color="Maroon" title="c4">h</font> is   <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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_6.html#D6" onmouseover="rs('group_6/D6')" onmouseout="rh()">GROUP_6:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:17"/>
 ex <font color="Olive" title="b1">h</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="c2">A</font>,<font color="Maroon" title="c3">B</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span>)</span>,<font color="Maroon" title="c1">G</font> st <br/>( <font color="Olive" title="b1">h</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="c1">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="c2">A</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="c3">B</font> holds <br/><font color="Olive" title="b1">h</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="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">b</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:17"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E11:17"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="txt" href="#E8:17"><span class="lab"><font color="Green" title="E23">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/>


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