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<span class="kw">let </span><font color="Maroon" title="c1">I</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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"> for <font color="Olive" title="b1">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font>  st ( for <font color="Olive" title="b2">w1</font>, <font color="Olive" title="b3">w2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">w2</font> holds <br/> <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b3">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Olive" title="b1">S</font> ) holds <br/><font color="Olive" title="b1">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L1" title="FSM_1:struct.1">FSM</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w2</font> holds <br/> <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> ) implies <font color="Maroon" title="c2">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
for <font color="Olive" title="b1">w1</font>, <font color="Olive" title="b2">w2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b1">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">w2</font> holds <br/> <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b2">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</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">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">w1</font>, <font color="Maroon" title="c4">w2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</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="c3">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:12_1"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c3">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><span class="kw">thus </span><a NAME="E2:12_1"/>
 <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:12_1_1_1"/>
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">I</font> or ( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_1_1"/>
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">I</font> or <font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">I</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:12_1_1_1_1"/>
 <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fsm_2.html#E8"><span class="lab"><font color="Green" title="E6">Lm1</font></span></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/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:12_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
( <font color="Maroon" title="c3">w1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">w2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> </span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c5">v1</font> = <font color="Maroon" title="c4">w2</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span></span>)</span>, <font color="Maroon" title="c6">v2</font> = <font color="Maroon" title="c3">w1</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span></span>)</span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T23" title="FINSEQ_1:th.23">FINSEQ_1:23</a></span>;<br/><a NAME="E3:12_1_1_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> )
 ;<br/><a NAME="E4:12_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A4</font></span>: 
( ( <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> ) or (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> &amp; <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T21" title="FINSEQ_1:th.21">FINSEQ_1:21</a>, <a class="ref" href="finseq_2.html#T23" title="FINSEQ_2:th.23">FINSEQ_2:23</a></span>;<br/><a NAME="E5:12_1_1_1_2"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:12_1_1_1_2"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E7:12_1_1_1_2"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">w2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:12_1_1_1_2"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="graph_2.html#T2" title="GRAPH_2:th.2">GRAPH_2:2</a></span>;<br/><a NAME="E8:12_1_1_1_2"/>
<font color="Maroon" title="c6">v2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:12_1_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:12_1_1_1_2"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="graph_2.html#T2" title="GRAPH_2:th.2">GRAPH_2:2</a></span>;<br/><a NAME="E9:12_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A8</font></span>: 
(  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> or  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E1:12_1"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E4:12_1_1_1_2"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E7:12_1_1_1_2"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">s1</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E11:12_1_1_1_2"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c4">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c7">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T101" title="FINSEQ_1:th.101">FINSEQ_1:101</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">s2</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E13:12_1_1_1_2"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c3">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c8">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T101" title="FINSEQ_1:th.101">FINSEQ_1:101</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c9">s1</font> = <font color="Maroon" title="c7">s1</font>, <font color="Maroon" title="c10">s2</font> = <font color="Maroon" title="c8">s2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_1_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E13:12_1_1_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></span>;<br/><a NAME="E15:12_1_1_1_2"/>
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:12_1_1_1_2_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="lab"><font color="Green" title="E20">A12</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">v1</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">w2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E3:12_1_1_1_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> )
 
;<br/>
<a NAME="E4:12_1_1_1_2_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span> or  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>;<br/>
<a NAME="E5:12_1_1_1_2_1"/><span class="kw">then </span>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_1_1_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E1:12_1_1_1_2_1"><span class="lab"><font color="Green" title="E19">A11</font></span></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/>


</div><span class="kw">end;</span></div><a NAME="E16:12_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E17:12_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A13</font></span>: 
 ex <font color="Olive" title="b1">WI</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ex <font color="Olive" title="b2">QI</font>, <font color="Olive" title="b3">QI1</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">WI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">v1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> &amp; <font color="Olive" title="b2">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> &amp; <font color="Olive" title="b3">QI1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">WI</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Olive" title="b2">QI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">QI1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><a NAME="E18:12_1_1_1_2"/>
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:12_1_1_1_2_2"/>
<font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:12_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<span class="lab"><font color="Green" title="E21">A15</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">v2</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">w1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E4:12_1_1_1_2_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font> )
 
;<br/>
<a NAME="E5:12_1_1_1_2_2"/><span class="kw">then </span>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">w2</font></span>)</span> or  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">v2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">w1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1_1_1_2_2"><span class="lab"><font color="Green" title="E21">A15</font></span></a>, <a class="ref" href="finseq_1.html#T9" title="FINSEQ_1:th.9">FINSEQ_1:9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:12_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12_1_1_1_2"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E2:12_1_1_1_2_2"><span class="lab"><font color="Green" title="E20">A14</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</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/>


</div><span class="kw">end;</span></div><a NAME="E19:12_1_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><a NAME="E20:12_1_1_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">WI2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> ex <font color="Olive" title="b2">QI2</font>, <font color="Olive" title="b3">QI12</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font> st <br/>( <font color="Olive" title="b1">WI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">v2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> &amp; <font color="Olive" title="b2">QI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> &amp; <font color="Olive" title="b3">QI12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">WI2</font> <a href="fsm_1.html#K1" title="FSM_1:func.1">-succ_of</a> <font color="Olive" title="b2">QI2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">QI12</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">q1</font> = <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <font color="Maroon" title="c12">q2</font> = <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> as   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:12_1_1_1_2"><span class="lab"><font color="Green" title="E19">A13</font></span></a></span>;<br/><a NAME="E22:12_1_1_1_2"/><span class="lab"><font color="Green" title="E20">A16</font></span>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c11">q1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><a NAME="E23:12_1_1_1_2"/><span class="lab"><font color="Green" title="E21">A17</font></span>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c10">s2</font>,<font color="Maroon" title="c12">q2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><a NAME="E24:12_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A18</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><a NAME="E25:12_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A19</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D2" title="FSM_1:def.2">FSM_1:def 2</a></span>;<br/><a NAME="E26:12_1_1_1_2"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c5">v1</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c11">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D3" title="FSM_1:def.3">FSM_1:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A20</font></span>:  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c11">q1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12_1_1_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="fsm_1.html#T23" title="FSM_1:th.23">FSM_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c5">v1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">s1</font>,<font color="Maroon" title="c11">q1</font></span>)</span>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:12_1_1_1_2"><span class="lab"><font color="Green" title="E20">A16</font></span></a>, <a class="txt" href="#E24:12_1_1_1_2"><span class="lab"><font color="Green" title="E22">A18</font></span></a>, <a class="txt" href="fsm_2.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>
;<br/><a NAME="E28:12_1_1_1_2"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c6">v2</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c12">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_1.html#D3" title="FSM_1:def.3">FSM_1:def 3</a></span>;<br/><span class="kw">then </span> <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> = 
<span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c10">s2</font>,<font color="Maroon" title="c12">q2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12_1_1_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="fsm_1.html#T23" title="FSM_1:th.23">FSM_1:23</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c6">v2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p2">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c10">s2</font>,<font color="Maroon" title="c12">q2</font></span>)</span>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:12_1_1_1_2"><span class="lab"><font color="Green" title="E21">A17</font></span></a>, <a class="txt" href="#E25:12_1_1_1_2"><span class="lab"><font color="Green" title="E23">A19</font></span></a>, <a class="txt" href="fsm_2.html#E9"><span class="lab"><font color="Green" title="E7">Lm2</font></span></a></span>
;<br/><a NAME="E30:12_1_1_1_2"/><span class="kw">then </span>
(  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> or  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:12_1_1_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E27:12_1_1_1_2"><span class="lab"><font color="Green" title="E24">A20</font></span></a>, <a class="ref" href="trees_1.html#T8" title="TREES_1:th.8">TREES_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E31:12_1_1_1_2"/>
 <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c3">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font>, <a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c4">w2</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c2">S</font> <a href="xboole_0.html#R3" title="XBOOLE_0:pred.3">are_c=-comparable</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D9" title="XBOOLE_0:def.9">XBOOLE_0:def 9</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/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:12"/>
<font color="Maroon" title="c2">S</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fsm_2.html#T5" target="_self" title="FSM_2:th.5">Th5</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/>


</div>
