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<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">IAlph</font>, <font color="Olive" title="b2">OAlph</font> being   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> <br/>  for <font color="Olive" title="b3">tfsm</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Olive" title="b1">IAlph</font>,<font color="Olive" title="b2">OAlph</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font> holds <br/> for <font color="Olive" title="b4">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b3">tfsm</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">IAlph</font>, <font color="Maroon" title="c3">OAlph</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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">tfsm</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c3">OAlph</font>  st <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font> holds <br/> for <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Olive" title="b1">tfsm</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">tfsm</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c2">IAlph</font>,<font color="Maroon" title="c3">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> implies  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:51"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:51"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font></span><br/>

<a NAME="E3:51"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
 ex <font color="Olive" title="b1">e</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">e</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:51"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:51"/>
 ex <font color="Olive" title="b1">d</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:51"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/>
<a NAME="E6:51"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> <a href="fsm_1.html#K7" title="FSM_1:func.7">-eq_states_partition</a> <font color="Maroon" title="c4">tfsm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="fsm_1.html#T29" target="_self" title="FSM_1:th.29">Th46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:51"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:51"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E3:51"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="fsm_1.html#T29" target="_self" title="FSM_1:th.29">Th46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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