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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y1</font>, <font color="Maroon" title="c3">y2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y1</font> &amp;  <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y2</font> implies <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:59"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y1</font>
 <span class="kw">and </span><br/><a NAME="E2:59"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="ringcat1.html#R2" title="RINGCAT1:pred.2">GO</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">y2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">a1</font>, <font color="Maroon" title="c5">a2</font>, <font color="Maroon" title="c6">a3</font>, <font color="Maroon" title="c7">a4</font>, <font color="Maroon" title="c8">a5</font>, <font color="Maroon" title="c9">a6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:59"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c7">a4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c8">a5</font>,<font color="Maroon" title="c9">a6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">and </span><br/><a NAME="E5:59"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 ex <font color="Olive" title="b1">G</font> being  <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>( <font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c7">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c8">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c9">a6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">Def16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">b2</font>, <font color="Maroon" title="c12">b3</font>, <font color="Maroon" title="c13">b4</font>, <font color="Maroon" title="c14">b5</font>, <font color="Maroon" title="c15">b6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:59"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>,<font color="Maroon" title="c13">b4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>,<font color="Maroon" title="c14">b5</font>,<font color="Maroon" title="c15">b6</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span>
 <span class="kw">and </span><br/><a NAME="E8:59"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
 ex <font color="Olive" title="b1">G</font> being  <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a> st <br/>( <font color="Maroon" title="c3">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c13">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c14">b5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Olive" title="b1">G</font> &amp; <font color="Maroon" title="c15">b6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Olive" title="b1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="ringcat1.html#D16" target="_self" title="RINGCAT1:def.16">Def16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">G2</font> being   <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><span class="kw"> such that </span><br/><a NAME="E10:59"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<font color="Maroon" title="c3">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">G2</font>
 <span class="kw">and </span><br/><a NAME="E11:59"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
( <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">G2</font> &amp; <font color="Maroon" title="c11">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c16">G2</font> )
 <span class="kw">and </span><br/><a NAME="E12:59"/>
<font color="Maroon" title="c12">b3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Maroon" title="c16">G2</font>
 <span class="kw">and </span><br/><a NAME="E13:59"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c13">b4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c16">G2</font>
 <span class="kw">and </span><br/><a NAME="E14:59"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
( <font color="Maroon" title="c14">b5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c16">G2</font> &amp; <font color="Maroon" title="c15">b6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c16">G2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59"><span class="lab"><font color="Green" title="E38">A6</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">G1</font> being   <a href="algstr_0.html#V36" title="ALGSTR_0:attr.36">strict</a>  <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a><span class="kw"> such that </span><br/><a NAME="E16:59"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">G1</font>
 <span class="kw">and </span><br/><a NAME="E17:59"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
( <font color="Maroon" title="c4">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c5">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c17">G1</font> )
 <span class="kw">and </span><br/><a NAME="E18:59"/>
<font color="Maroon" title="c6">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_1.html#K5" title="VECTSP_1:func.5">comp</a> <font color="Maroon" title="c17">G1</font>
 <span class="kw">and </span><br/><a NAME="E19:59"/><span class="lab"><font color="Green" title="E45">A13</font></span>: 
<font color="Maroon" title="c7">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c17">G1</font>
 <span class="kw">and </span><br/><a NAME="E20:59"/><span class="lab"><font color="Green" title="E46">A14</font></span>: 
( <font color="Maroon" title="c8">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c17">G1</font> &amp; <font color="Maroon" title="c9">a6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c17">G1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E36">A4</font></span></a></span>;<br/>
<a NAME="E21:59"/><span class="lab"><font color="Green" title="E47">A15</font></span>: 
( the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c17">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U2" title="ALGSTR_0:sel.2">multF</a> of <font color="Maroon" title="c16">G2</font> &amp;  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c17">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c16">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E20:59"><span class="lab"><font color="Green" title="E46">A14</font></span></a>, <a class="txt" href="#E14:59"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E22:59"/><span class="lab"><font color="Green" title="E48">A16</font></span>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c4">a1</font>,<font color="Maroon" title="c5">a2</font>,<font color="Maroon" title="c6">a3</font>,<font color="Maroon" title="c7">a4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c10">b1</font>,<font color="Maroon" title="c11">b2</font>,<font color="Maroon" title="c12">b3</font>,<font color="Maroon" title="c13">b4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="txt" href="#E7:59"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="mcart_1.html#T28" title="MCART_1:th.28">MCART_1:28</a></span>;<br/>
<a NAME="E23:59"/><span class="kw">then </span>
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c17">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c16">G2</font> &amp; the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c17">G1</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> the <a href="algstr_0.html#U1" title="ALGSTR_0:sel.1">addF</a> of <font color="Maroon" title="c16">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="txt" href="#E11:59"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></span>;<br/>
<span class="kw">hence </span><a NAME="E24:59"/>
<font color="Maroon" title="c2">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:59"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E19:59"><span class="lab"><font color="Green" title="E45">A13</font></span></a>, <a class="txt" href="#E10:59"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="txt" href="#E13:59"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E22:59"><span class="lab"><font color="Green" title="E48">A16</font></span></a>, <a class="txt" href="#E21:59"><span class="lab"><font color="Green" title="E47">A15</font></span></a>, <a class="ref" href="mcart_1.html#T33" title="MCART_1:th.33">MCART_1:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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