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<span class="kw">let </span><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">O</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   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">M</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_2.html#L2" title="FSM_2:struct.2">Moore-SM_Final</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">O</font>  st <font color="Olive" title="b2">M</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Olive" title="b1">s</font> <a href="fsm_2.html#R2" title="FSM_2:pred.2">leads_to_final_state_of</a> <font color="Olive" title="b2">M</font> holds <br/> ex <font color="Olive" title="b3">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">O</font> st <font color="Olive" title="b3">t</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">M</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_2.html#L2" title="FSM_2:struct.2">Moore-SM_Final</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">O</font>  st <font color="Olive" title="b1">M</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="fsm_2.html#R2" title="FSM_2:pred.2">leads_to_final_state_of</a> <font color="Olive" title="b1">M</font> holds <br/> ex <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">O</font> st <font color="Olive" title="b2">t</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <font color="Maroon" title="c3">s</font>,<font color="Olive" title="b1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">M</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_2.html#L2" title="FSM_2:struct.2">Moore-SM_Final</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">O</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="c4">M</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a>  &amp; <font color="Maroon" title="c3">s</font> <a href="fsm_2.html#R2" title="FSM_2:pred.2">leads_to_final_state_of</a> <font color="Maroon" title="c4">M</font> implies  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">O</font> st <font color="Olive" title="b1">t</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">M</font> )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:38"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c4">M</font> is  <a href="fsm_2.html#V1" title="FSM_2:attr.1">calculating_type</a> 
 <span class="kw">and </span><br/><a NAME="E2:38"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">s</font> <a href="fsm_2.html#R2" title="FSM_2:pred.2">leads_to_final_state_of</a> <font color="Maroon" title="c4">M</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">O</font> st <font color="Olive" title="b1">t</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">M</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">q</font> being   <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <font color="Maroon" title="c4">M</font><span class="kw"> such that </span><br/><a NAME="E4:38"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="fsm_2.html#R1" title="FSM_2:pred.1">is_accessible_via</a> <font color="Maroon" title="c3">s</font>
 <span class="kw">and </span><br/><a NAME="E5:38"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="fsm_2.html#D5" target="_self" title="FSM_2:def.5">Def5</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font><span class="kw"> such that </span><br/><a NAME="E7:38"/><span class="lab"><font color="Green" title="E24">A5</font></span>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c5">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:38"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="fsm_2.html#D2" target="_self" title="FSM_2:def.2">Def2</a></span>;<br/>
<a NAME="E8:38"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</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> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:38"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="fsm_1.html#D3" title="FSM_1:def.3">FSM_1:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">m</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E10:38"/><span class="lab"><font color="Green" title="E26">A7</font></span>: 
for <font color="Olive" title="b1">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font>  st <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">w</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">s</font> holds <br/><span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E11:38"/><span class="lab"><font color="Green" title="E27">A8</font></span>: 
for <font color="Olive" title="b1">w</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">j</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b1">w</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">s</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E2:38"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="fsm_2.html#T18" target="_self" title="FSM_2:th.18">Th18</a></span>;<br/>
<a NAME="E12:38"/><span class="lab"><font color="Green" title="E28">A9</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</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="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>;<br/>
<a NAME="E13:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E8:38"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="txt" href="#E11:38"><span class="lab"><font color="Green" title="E27">A8</font></span></a></span>;<br/>
<a NAME="E14:38"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E12:38"><span class="lab"><font color="Green" title="E28">A9</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">t</font> = the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c4">M</font> <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="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <span class="p3">(<span class="default"><span class="p4"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</font></span>)</span>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">O</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">t</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="c8">t</font> <a href="fsm_2.html#R3" title="FSM_2:pred.3">is_result_of</a> <font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c4">M</font></span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c7">m</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="fsm_2.html#D8" target="_self" title="FSM_2:def.8">FSM_2:def 8</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">w</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">w</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">s</font> holds <br/>( ( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c4">M</font> <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="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) ) &amp; ( for <font color="Olive" title="b2">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Olive" title="b1">w</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">w1</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="c9">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="c3">s</font> implies ( ( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c4">M</font> <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="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E16:38"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<font color="Maroon" title="c9">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="c3">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="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c4">M</font> <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="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) ) &amp; ( for <font color="Olive" title="b1">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) )</span><br/>

<a NAME="E17:38"/>
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">s</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">w</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="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E18:38"/>
( <font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <font color="Maroon" title="c8">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="fsm_1.html#U4" title="FSM_1:sel.4">OFun</a> of <font color="Maroon" title="c4">M</font> <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="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:38"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E10:38"><span class="lab"><font color="Green" title="E26">A7</font></span></a>, <a class="txt" href="#E13:38"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="fsm_2.html#D1" target="_self" title="FSM_2:def.1">Def1</a></span>; <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">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font></span><br/>

<span class="kw">thus </span><a NAME="E19:38"/>
for <font color="Olive" title="b1">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">w1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/> not <span class="p1">(<span class="default"><a href="fsm_2.html#NK1" title="FSM_2:NK.1">GEN</a> <font color="Maroon" title="c9">w1</font>,the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">M</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="fsm_2.html#U1" title="FSM_2:sel.1">FinalS</a> of <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:38"><span class="lab"><font color="Green" title="E27">A8</font></span></a>, <a class="txt" href="#E16:38"><span class="lab"><font color="Green" title="E30">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>
