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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lang1.html#V1" title="LANG1:attr.1">strict</a>  <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">p</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>] ) ) &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">p</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>] ) ) implies <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:15_1"/><i><font color="Green" title="E8">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">p</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>] ) ) )
 <b>and </b><br/><a NAME="E2:15_1"/><i><font color="Green" title="E9">A2</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">x</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">p</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>] ) ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font> ) &amp; ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font> ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:15_1_1_1"/><i><font color="Green" title="E10">A3</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font></span><br/><a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green" title="E11">A4</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="dtconstr.html#E2:15_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="finseq_2.html#D3" title="FINSEQ_2:def.3">FINSEQ_2:def 3</a></i>;<br/><a NAME="E5:15_1_1_1"/><i><font color="Green" title="E12">A5</font></i>: 
( <font color="Maroon" title="c5">x1</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c6">y1</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">y1</font>] )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">x2</font> = <font color="Maroon" title="c3">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E1:15_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y2</font> = <font color="Maroon" title="c4">y</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1_1_1"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="finseq_2.html#D3" title="FINSEQ_2:def.3">FINSEQ_2:def 3</a></i>;<br/><a NAME="E8:15_1_1_1"/>
<font color="Maroon" title="c7">x2</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c8">y2</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E1:15_1_1_1"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="dtconstr.html#E5:15_1_1_1"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:15_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:15_1_1"/><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font></span><br/><a NAME="E3:15_1_1"/><b>then </b><i><font color="Green" title="E11">A7</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">x2</font> = <font color="Maroon" title="c3">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="dtconstr.html#E2:15_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c4">y</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="dtconstr.html#E3:15_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="finseq_2.html#D3" title="FINSEQ_2:def.3">FINSEQ_2:def 3</a></i>;<br/><a NAME="E6:15_1_1"/><i><font color="Green" title="E12">A8</font></i>: 
( <font color="Maroon" title="c5">x2</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c6">y2</font> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">y2</font>] )
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c4">y</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E3:15_1_1"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="finseq_2.html#D3" title="FINSEQ_2:def.3">FINSEQ_2:def 3</a></i>;<br/><a NAME="E9:15_1_1"/>
<font color="Maroon" title="c7">x1</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c8">y1</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1_1"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="dtconstr.html#E6:15_1_1"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/><b>hence </b><a NAME="E10:15_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">G1</font>
 <b>by </b><i><a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon" title="c1">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:15_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E2:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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