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<b>let </b><font color="Maroon" title="c1">IAlph</font>, <font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></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="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font><br/> for <font color="Olive" title="b2">CRtfsm1</font>, <font color="Olive" title="b3">CRtfsm2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">CRtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">CRtfsm2</font> &amp; <font color="Olive" title="b2">CRtfsm1</font>,<font color="Olive" title="b3">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Olive" title="b1">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Olive" title="b3">CRtfsm2</font> holds <br/>for <font color="Olive" title="b4">Q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Olive" title="b1">tfsm</font></span>)</span>  ex <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b4">Q</font> st <br/>( <font color="Olive" title="b5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">CRtfsm1</font> &amp; <font color="Olive" title="b6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b3">CRtfsm2</font> &amp; ( for <font color="Olive" title="b7">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b4">Q</font>  holds <br/>( <font color="Olive" title="b7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">q1</font> or <font color="Olive" title="b7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">q2</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c3">tfsm</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">CRtfsm1</font>, <font color="Olive" title="b2">CRtfsm2</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>  st the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">CRtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">CRtfsm2</font> &amp; <font color="Olive" title="b1">CRtfsm1</font>,<font color="Olive" title="b2">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Olive" title="b2">CRtfsm2</font> holds <br/>for <font color="Olive" title="b3">Q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>  ex <font color="Olive" title="b4">q1</font>, <font color="Olive" title="b5">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">Q</font> st <br/>( <font color="Olive" title="b4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">CRtfsm1</font> &amp; <font color="Olive" title="b5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">CRtfsm2</font> &amp; ( for <font color="Olive" title="b6">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b3">Q</font>  holds <br/>( <font color="Olive" title="b6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q1</font> or <font color="Olive" title="b6">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">q2</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c4">CRtfsm1</font>, <font color="Maroon" title="c5">CRtfsm2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> <a href="fsm_1.html#V5" title="FSM_1:attr.5">reduced</a> <a href="fsm_1.html#V7" title="FSM_1:attr.7">connected</a>  <a href="fsm_1.html#L2" title="FSM_1:struct.2">Mealy-FSM</a> of <font color="Maroon" title="c1">IAlph</font>,<font color="Maroon" title="c2">OAlph</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> &amp; <font color="Maroon" title="c4">CRtfsm1</font>,<font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Maroon" title="c5">CRtfsm2</font> implies for <font color="Olive" title="b1">Q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>  ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font> st <br/>( <font color="Olive" title="b2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> &amp; ( for <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font>  holds <br/>( <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q1</font> or <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q2</font> ) ) ) )</span><br/>









<b>set </b><font color="Maroon" title="c6">rtfsm1</font> = <font color="Maroon" title="c4">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">rtfsm2</font> = <font color="Maroon" title="c5">CRtfsm2</font>;<br/>
<b>assume </b><a NAME="E1:110"/><i><font color="Green" title="E81">A1</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> &amp; <font color="Maroon" title="c4">CRtfsm1</font>,<font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#R4" title="FSM_1:pred.4">-are_equivalent</a>  &amp; <font color="Maroon" title="c3">tfsm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K13" title="FSM_1:func.13">-Mealy_union</a> <font color="Maroon" title="c5">CRtfsm2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">Q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>  ex <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font> st <br/>( <font color="Olive" title="b2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> &amp; <font color="Olive" title="b3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> &amp; ( for <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font>  holds <br/>( <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q1</font> or <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q2</font> ) ) )</span><br/>

<b>assume </b><a NAME="E2:110"/>
 ex <font color="Olive" title="b1">Q</font> being  <a href="fsm_1.html#NM1" title="FSM_1:NM.1">State</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> st <br/>for <font color="Olive" title="b2">q1</font>, <font color="Olive" title="b3">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font>  holds <br/>(  not <font color="Olive" title="b2">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Olive" title="b3">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  ex <font color="Olive" title="b4">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Olive" title="b1">Q</font> st <br/>( not <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q1</font> &amp; not <font color="Olive" title="b4">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c8">Q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span><b> such that </b><br/><a NAME="E4:110"/><i><font color="Green" title="E82">A2</font></i>: 
for <font color="Olive" title="b1">q1</font>, <font color="Olive" title="b2">q2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Q</font>  holds <br/>(  not <font color="Olive" title="b1">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Olive" title="b2">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  ex <font color="Olive" title="b3">q</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c8">Q</font> st <br/>( not <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q1</font> &amp; not <font color="Olive" title="b3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q2</font> ) )
 ;<br/>
<a NAME="E5:110"/><i><font color="Green" title="E83">A3</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">rtfsm</font> =  <a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c10">Srtfsm1</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c11">ISrtfsm1</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font>;<br/>
<b>set </b><font color="Maroon" title="c12">Srtfsm2</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>;<br/>
<b>set </b><font color="Maroon" title="c13">ISrtfsm2</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font>;<br/>
<b>set </b><font color="Maroon" title="c14">Stfsm</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>;<br/>
<b>set </b><font color="Maroon" title="c15">Srtfsm</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>;<br/>
<a NAME="E6:110"/><i><font color="Green" title="E84">A4</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">Q</font> = <font color="Maroon" title="c8">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:110"/>
<font color="Maroon" title="c16">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 
;<br/>
<a NAME="E9:110"/><b>then </b><i><font color="Green" title="E85">A5</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<a NAME="E10:110"/><b>then </b><i><font color="Green" title="E86">A6</font></i>: 
<font color="Maroon" title="c16">Q</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> 
 
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E12:110"/><i><font color="Green" title="E87">A7</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
 <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <span class="p1">(<span class="default"><a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> = 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font>
<b>by </b><i><a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
<b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>
;<br/>
<a NAME="E14:110"/><b>then </b><i><font color="Green" title="E88">A8</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 
<b>by </b><i><a class="txt" href="#E6:110"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2"/>
( <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> )
 </span><b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="#E14:110"><i><font color="Green" title="E88">A8</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1"><b>suppose </b></a><a NAME="E1:110_2_1"/><i><font color="Green" title="E89">A9</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c18">Q'</font> = <font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><div><i><font color="Green" title="E90">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_1"/><i><font color="Green" title="E91">A11</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <b>by </b><i><a class="txt" href="#E1:110_2_1"><i><font color="Green" title="E89">A9</font></i></a></i>;<br/><a NAME="E3:110_2_1_1"/>
<font color="Maroon" title="c19">q</font> is <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E5:110_2_1_1"/><i><font color="Green" title="E92">A12</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c19">q</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></i>;<br/><b>set </b><font color="Maroon" title="c21">adl</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font>;<br/><div><i><font color="Green" title="E93">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_1_1"/><i><font color="Green" title="E94">A14</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_1_1_1"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <b>by </b><i><a class="txt" href="#E5:110"><i><font color="Green" title="E83">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E3:110_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1_1"><i><font color="Green" title="E91">A11</font></i></a>, <a class="txt" href="#E1:110_2_1_1_1"><i><font color="Green" title="E94">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_1_2"/><i><font color="Green" title="E94">A15</font></i>: 
not the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_1_1_2"/><i><font color="Green" title="E95">A16</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c19">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c23">adl</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E2:110_2_1_1_2"><i><font color="Green" title="E95">A16</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/><a NAME="E5:110_2_1_1_2"/><i><font color="Green" title="E96">A17</font></i>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></i>;<br/><a NAME="E6:110_2_1_1_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> or  not <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_2_1_1_2_1"/>
 ex <font color="Olive" title="b1">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> st <br/>( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c24">Y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span><b> such that </b><br/><a NAME="E3:110_2_1_1_2_1"/><i><font color="Green" title="E97">A18</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Y</font> &amp; <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Y</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c25">Y</font> = <font color="Maroon" title="c24">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="#E6:110"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">Q</font> = <font color="Maroon" title="c16">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/>
<a NAME="E6:110_2_1_1_2_1"/>
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 
;<br/>
<a NAME="E7:110_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<a NAME="E8:110_2_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c25">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c26">Q</font>
 
<b>by </b><i><a class="txt" href="#E9:110"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="#E1:110_2_1_1_2"><i><font color="Green" title="E94">A15</font></i></a>, <a class="txt" href="#E3:110_2_1_1_2_1"><i><font color="Green" title="E97">A18</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E9:110_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="#E3:110_2_1_1_2_1"><i><font color="Green" title="E97">A18</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:110_2_1_1_2"/><b>then </b>
not <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c23">adl</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <b>by </b><i><a class="txt" href="#E6:110"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="#E5:110_2_1_1_2"><i><font color="Green" title="E96">A17</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">dseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E9:110_2_1_1_2"/><i><font color="Green" title="E97">A19</font></i>: 
<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">adl</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">adl2</font> = <font color="Maroon" title="c23">adl</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> ;<br/><b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c22">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> ;<br/><a NAME="E12:110_2_1_1_2"/><i><font color="Green" title="E98">A20</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c25">adl2</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D5" target="_self" title="FSM_1:def.5">Def5</a></i>;<br/><a NAME="E13:110_2_1_1_2"/>
<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T72" target="_self" title="FSM_1:th.72">Th72</a></i>;<br/><a NAME="E14:110_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">adl2</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E9:110_2_1_1_2"><i><font color="Green" title="E97">A19</font></i></a>, <a class="ref" href="fsm_1.html#T74" target="_self" title="FSM_1:th.74">Th74</a></i>;<br/><a NAME="E15:110_2_1_1_2"/><b>then </b>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">dseq</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">dseq</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E5:110_2_1_1"><i><font color="Green" title="E92">A12</font></i></a>, <a class="txt" href="#E12:110_2_1_1_2"><i><font color="Green" title="E98">A20</font></i></a>, <a class="ref" href="fsm_1.html#T27" target="_self" title="FSM_1:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E16:110_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#D9" target="_self" title="FSM_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:110_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:110_2_1_1"><i><font color="Green" title="E93">A13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_2"/><i><font color="Green" title="E91">A21</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_1_2"/><i><font color="Green" title="E92">A22</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">Q</font> = <font color="Maroon" title="c16">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c20">Q'</font> = <font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c21">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E6:110_2_1_2"/><i><font color="Green" title="E93">A23</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Q'</font>
 <b>by </b><i><a class="txt" href="#E1:110_2_1_2"><i><font color="Green" title="E91">A21</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E7:110_2_1_2"/><i><font color="Green" title="E94">A24</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="txt" href="#E14:110"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="#E2:110_2_1_2"><i><font color="Green" title="E92">A22</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2_1_2_1"/>
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> )
 </span><b>by </b><i><a class="txt" href="#E6:110_2_1_2"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E7:110_2_1_2"><i><font color="Green" title="E94">A24</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:110_2_1_2_1_1"/><i><font color="Green" title="E95">A25</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_1_2"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:110_2_1_2_1_1"/>
(  not <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self" title="FSM_1:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E5:110_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="#E6:110_2_1_2"><i><font color="Green" title="E93">A23</font></i></a>, <a class="txt" href="#E1:110_2_1_2_1_1"><i><font color="Green" title="E95">A25</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:110_2_1_2_1_2"/><i><font color="Green" title="E95">A26</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c22">Q''</font> = <font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><div><i><font color="Green" title="E96">A27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_2_1_2_1"/><i><font color="Green" title="E97">A28</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_1_2_1_2_1"/><i><font color="Green" title="E98">A29</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font>  holds <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_2_1_2_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c23">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font><b> such that </b><a NAME="E2:110_2_1_2_1_2_1_1"/><i><font color="Green" title="E99">A30</font></i>: 
( <font color="Maroon" title="c23">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">q</font> &amp; <font color="Maroon" title="c23">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:110_2_1_2_1_2_1_1"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Q'</font>
 
<b>by </b><i><a class="txt" href="#E10:110"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="#E2:110_2_1_2_1_2_1_1"><i><font color="Green" title="E99">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E4:110_2_1_2_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1_2_1_2_1"><i><font color="Green" title="E97">A28</font></i></a>, <a class="txt" href="#E2:110_2_1_2_1_2_1_1"><i><font color="Green" title="E99">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c23">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_1_2"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_1_2_1_2_1"/>
(  not <font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">q</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">x</font> ) )
 <b>by </b><i><a class="txt" href="#E4:110"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:110_2_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="#E1:110_2_1_2_1_2"><i><font color="Green" title="E95">A26</font></i></a>, <a class="txt" href="#E2:110_2_1_2_1_2_1"><i><font color="Green" title="E98">A29</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_1_2_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_1_2_1_2_2"/><i><font color="Green" title="E97">A31</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_1_2_1_2_2"/><i><font color="Green" title="E98">A32</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Q'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">Q'</font> = <font color="Maroon" title="c20">Q'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c24">Q''</font> = <font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c25">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E6:110_2_1_2_1_2_2"/><i><font color="Green" title="E99">A33</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Q''</font>
 <b>by </b><i><a class="txt" href="#E1:110_2_1_2_1_2_2"><i><font color="Green" title="E97">A31</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E7:110_2_1_2_1_2_2"/><i><font color="Green" title="E100">A34</font></i>: 
<font color="Maroon" title="c24">Q''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="txt" href="#E7:110_2_1_2"><i><font color="Green" title="E94">A24</font></i></a>, <a class="txt" href="#E2:110_2_1_2_1_2_2"><i><font color="Green" title="E98">A32</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2_1_2_1_2_2_1"/>
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> )
 </span><b>by </b><i><a class="txt" href="#E6:110_2_1_2_1_2_2"><i><font color="Green" title="E99">A33</font></i></a>, <a class="txt" href="#E7:110_2_1_2_1_2_2"><i><font color="Green" title="E100">A34</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:110_2_1_2_1_2_2_1_1"/><i><font color="Green" title="E101">A35</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:110_2_1_2_1_2_2_1_1"/><i><font color="Green" title="E102">A36</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q'</font>
 <b>by </b><i><a class="txt" href="#E6:110_2_1_2_1_2_2"><i><font color="Green" title="E99">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c25">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E2:110_2_1_2_1_2_2_1_1"><i><font color="Green" title="E102">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_1_2_1_2_2_1_1"/>
(  not <font color="Maroon" title="c26">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c26">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self" title="FSM_1:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E6:110_2_1_2_1_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1"><i><font color="Green" title="E89">A9</font></i></a>, <a class="txt" href="#E1:110_2_1_2_1_2_2_1_1"><i><font color="Green" title="E101">A35</font></i></a>, <a class="txt" href="#E2:110_2_1_2_1_2_2_1_1"><i><font color="Green" title="E102">A36</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_1_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:110_2_1_2_1_2_2_1_2"/><i><font color="Green" title="E101">A37</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_1_2"><i><font color="Green" title="E93">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E3:110_2_1_2_1_2_2_1_2"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q'</font>
 <b>by </b><i><a class="txt" href="#E6:110_2_1_2_1_2_2"><i><font color="Green" title="E99">A33</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c25">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_1_2_1_2_2_1_2"/>
(  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self" title="FSM_1:th.79">Th79</a></i>;<br/><b>hence </b><a NAME="E6:110_2_1_2_1_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_1_2_1_2"><i><font color="Green" title="E95">A26</font></i></a>, <a class="txt" href="#E6:110_2_1_2_1_2_2"><i><font color="Green" title="E99">A33</font></i></a>, <a class="txt" href="#E1:110_2_1_2_1_2_2_1_2"><i><font color="Green" title="E101">A37</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:110_2_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:110_2_1_2_1_2"><i><font color="Green" title="E96">A27</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:110_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:110_2_1"><i><font color="Green" title="E90">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2"><b>suppose </b></a><a NAME="E1:110_2_2"/><i><font color="Green" title="E89">A38</font></i>: 
<font color="Maroon" title="c17">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c18">Q'</font> = <font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><div><i><font color="Green" title="E90">A39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_1"/><i><font color="Green" title="E91">A40</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>reconsider </b><font color="Maroon" title="c19">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> <b>by </b><i><a class="txt" href="#E1:110_2_2"><i><font color="Green" title="E89">A38</font></i></a></i>;<br/><a NAME="E3:110_2_2_1"/>
<font color="Maroon" title="c19">q</font> is <a href="fsm_1.html#V6" title="FSM_1:attr.6">accessible</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D22" target="_self" title="FSM_1:def.22">Def22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c20">w</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E5:110_2_2_1"/><i><font color="Green" title="E92">A41</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c19">q</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D21" target="_self" title="FSM_1:def.21">Def21</a></i>;<br/><b>set </b><font color="Maroon" title="c21">adl</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font>;<br/><div><i><font color="Green" title="E93">A42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_1_1"/><i><font color="Green" title="E94">A43</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_2_1_1"/>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q</font>
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>hence </b><a NAME="E3:110_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2_1"><i><font color="Green" title="E91">A40</font></i></a>, <a class="txt" href="#E1:110_2_2_1_1"><i><font color="Green" title="E94">A43</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_1_2"/><i><font color="Green" title="E94">A44</font></i>: 
not the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_2_1_2"/><i><font color="Green" title="E95">A45</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c19">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c23">adl</font> = the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="fsm_1.html#K3" title="FSM_1:func.3">leads_to_under</a> <font color="Maroon" title="c20">w</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E2:110_2_2_1_2"><i><font color="Green" title="E95">A45</font></i></a>, <a class="ref" href="fsm_1.html#D24" target="_self" title="FSM_1:def.24">Def24</a></i>;<br/><a NAME="E5:110_2_2_1_2"/><i><font color="Green" title="E96">A46</font></i>: 
 <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font> is <a href="fsm_1.html#V4" title="FSM_1:attr.4">final</a>
 <b>by </b><i><a class="ref" href="fsm_1.html#D15" target="_self" title="FSM_1:def.15">Def15</a></i>;<br/><a NAME="E6:110_2_2_1_2"/>
for <font color="Olive" title="b1">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> or  not <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:110_2_2_1_2_1"/>
 ex <font color="Olive" title="b1">Y</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span> st <br/>( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> &amp; <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>then consider </b><font color="Maroon" title="c24">Y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span><b> such that </b><br/><a NAME="E3:110_2_2_1_2_1"/><i><font color="Green" title="E97">A47</font></i>: 
( <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Y</font> &amp; <font color="Maroon" title="c23">adl</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Y</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c25">Y</font> = <font color="Maroon" title="c24">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> <b>by </b><i><a class="txt" href="#E6:110"><i><font color="Green" title="E84">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:110_2_2_1_2_1"/>
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fsm_1.html#K11" title="FSM_1:func.11">the_reduction_of</a> <font color="Maroon" title="c3">tfsm</font></span>)</span>
 
;<br/>
<a NAME="E6:110_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c25">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fsm_1.html#K8" title="FSM_1:func.8">final_states_partition</a> <font color="Maroon" title="c3">tfsm</font>
 
<b>by </b><i><a class="ref" href="fsm_1.html#D18" target="_self" title="FSM_1:def.18">Def18</a></i>;<br/>
<a NAME="E7:110_2_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c25">Y</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c16">Q</font>
 
<b>by </b><i><a class="txt" href="#E9:110"><i><font color="Green" title="E85">A5</font></i></a>, <a class="txt" href="#E1:110_2_2_1_2"><i><font color="Green" title="E94">A44</font></i></a>, <a class="txt" href="#E3:110_2_2_1_2_1"><i><font color="Green" title="E97">A47</font></i></a>, <a class="ref" href="eqrel_1.html#D6" title="EQREL_1:def.6">EQREL_1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E8:110_2_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a>, <a class="txt" href="#E3:110_2_2_1_2_1"><i><font color="Green" title="E97">A47</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E7:110_2_2_1_2"/><b>then </b>
not <font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c23">adl</font> <a href="fsm_1.html#R5" title="FSM_1:pred.5">-are_equivalent</a> 
 <b>by </b><i><a class="txt" href="#E6:110"><i><font color="Green" title="E84">A4</font></i></a>, <a class="txt" href="#E5:110_2_2_1_2"><i><font color="Green" title="E96">A46</font></i></a>, <a class="ref" href="fsm_1.html#D14" target="_self" title="FSM_1:def.14">Def14</a></i>;<br/><b>then consider </b><font color="Maroon" title="c24">dseq</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">IAlph</font><b> such that </b><br/><a NAME="E9:110_2_2_1_2"/><i><font color="Green" title="E97">A48</font></i>: 
<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">adl</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="ref" href="fsm_1.html#D10" target="_self" title="FSM_1:def.10">Def10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c25">adl2</font> = <font color="Maroon" title="c23">adl</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> ;<br/><b>reconsider </b><font color="Maroon" title="c26">q1</font> = <font color="Maroon" title="c22">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> ;<br/><a NAME="E12:110_2_2_1_2"/><i><font color="Green" title="E98">A49</font></i>: 
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font>,<font color="Maroon" title="c20">w</font> <a href="fsm_1.html#R1" title="FSM_1:pred.1">-leads_to</a> <font color="Maroon" title="c25">adl2</font>
 <b>by </b><i><a class="ref" href="fsm_1.html#D5" target="_self" title="FSM_1:def.5">Def5</a></i>;<br/><a NAME="E13:110_2_2_1_2"/>
<font color="Maroon" title="c22">q</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T74" target="_self" title="FSM_1:th.74">Th74</a></i>;<br/><a NAME="E14:110_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c26">q1</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c25">adl2</font>,<font color="Maroon" title="c24">dseq</font> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="txt" href="#E9:110_2_2_1_2"><i><font color="Green" title="E97">A48</font></i></a>, <a class="ref" href="fsm_1.html#T72" target="_self" title="FSM_1:th.72">Th72</a></i>;<br/><a NAME="E15:110_2_2_1_2"/><b>then </b>
the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c5">CRtfsm2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">dseq</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="fsm_1.html#U2" title="FSM_1:sel.2">InitS</a> of <font color="Maroon" title="c4">CRtfsm1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">w</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c24">dseq</font></span>)</span> <a href="fsm_1.html#K4" title="FSM_1:func.4">-response</a> 
 <b>by </b><i><a class="txt" href="#E5:110_2_2_1"><i><font color="Green" title="E92">A41</font></i></a>, <a class="txt" href="#E12:110_2_2_1_2"><i><font color="Green" title="E98">A49</font></i></a>, <a class="ref" href="fsm_1.html#T27" target="_self" title="FSM_1:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E16:110_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#D9" target="_self" title="FSM_1:def.9">Def9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E8:110_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:110_2_2_1"><i><font color="Green" title="E93">A42</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_2"/><i><font color="Green" title="E91">A50</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_2_2"/><i><font color="Green" title="E92">A51</font></i>: 
<font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c19">Q</font> = <font color="Maroon" title="c16">Q</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c20">Q'</font> = <font color="Maroon" title="c16">Q</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c17">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c21">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E6:110_2_2_2"/><i><font color="Green" title="E93">A52</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Q'</font>
 <b>by </b><i><a class="txt" href="#E1:110_2_2_2"><i><font color="Green" title="E91">A50</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E7:110_2_2_2"/><i><font color="Green" title="E94">A53</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="txt" href="#E14:110"><i><font color="Green" title="E88">A8</font></i></a>, <a class="txt" href="#E2:110_2_2_2"><i><font color="Green" title="E92">A51</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2_2_2_1"/>
( <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> )
 </span><b>by </b><i><a class="txt" href="#E6:110_2_2_2"><i><font color="Green" title="E93">A52</font></i></a>, <a class="txt" href="#E7:110_2_2_2"><i><font color="Green" title="E94">A53</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1_1"><b>suppose </b></a><a NAME="E1:110_2_2_2_1_1"/><i><font color="Green" title="E95">A54</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>set </b><font color="Maroon" title="c22">Q''</font> = <font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/><div><i><font color="Green" title="E96">A55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_2_1_1_1"/><i><font color="Green" title="E97">A56</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_2_2_1_1_1"/><i><font color="Green" title="E98">A57</font></i>: 
for <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font>  holds <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">q</font> or <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c23">z</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font><b> such that </b><a NAME="E2:110_2_2_2_1_1_1_1"/><i><font color="Green" title="E99">A58</font></i>: 
( <font color="Maroon" title="c23">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">q</font> &amp; <font color="Maroon" title="c23">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c21">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:110_2_2_2_1_1_1_1"/>
<font color="Maroon" title="c23">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c20">Q'</font>
 
<b>by </b><i><a class="txt" href="#E10:110"><i><font color="Green" title="E86">A6</font></i></a>, <a class="txt" href="#E2:110_2_2_2_1_1_1_1"><i><font color="Green" title="E99">A58</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/>
<b>hence </b><a NAME="E4:110_2_2_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2_2_1_1_1"><i><font color="Green" title="E97">A56</font></i></a>, <a class="txt" href="#E2:110_2_2_2_1_1_1_1"><i><font color="Green" title="E99">A58</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c23">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c24">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_2_2"><i><font color="Green" title="E93">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_2_2_1_1_1"/>
(  not <font color="Maroon" title="c24">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Maroon" title="c23">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  ex <font color="Olive" title="b1">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> st <br/>( <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c24">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">q</font> ) )
 <b>by </b><i><a class="txt" href="#E4:110"><i><font color="Green" title="E82">A2</font></i></a></i>;<br/><b>hence </b><a NAME="E6:110_2_2_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2"><i><font color="Green" title="E89">A38</font></i></a>, <a class="txt" href="#E1:110_2_2_2_1_1"><i><font color="Green" title="E95">A54</font></i></a>, <a class="txt" href="#E2:110_2_2_2_1_1_1"><i><font color="Green" title="E98">A57</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="110_2_2_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:110_2_2_2_1_1_2"/><i><font color="Green" title="E97">A59</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:110_2_2_2_1_1_2"/><i><font color="Green" title="E98">A60</font></i>: 
<font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Q'</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">Q'</font> = <font color="Maroon" title="c20">Q'</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>reconsider </b><font color="Maroon" title="c24">Q''</font> = <font color="Maroon" title="c20">Q'</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c21">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font> ;<br/><b>consider </b><font color="Maroon" title="c25">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">tfsm</font><b> such that </b><br/><a NAME="E6:110_2_2_2_1_1_2"/><i><font color="Green" title="E99">A61</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c24">Q''</font>
 <b>by </b><i><a class="txt" href="#E1:110_2_2_2_1_1_2"><i><font color="Green" title="E97">A59</font></i></a>, <a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><a NAME="E7:110_2_2_2_1_1_2"/><i><font color="Green" title="E100">A62</font></i>: 
<font color="Maroon" title="c24">Q''</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 <b>by </b><i><a class="txt" href="#E7:110_2_2_2"><i><font color="Green" title="E94">A53</font></i></a>, <a class="txt" href="#E2:110_2_2_2_1_1_2"><i><font color="Green" title="E98">A60</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:110_2_2_2_1_1_2_1"/>
( <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> )
 </span><b>by </b><i><a class="txt" href="#E6:110_2_2_2_1_1_2"><i><font color="Green" title="E99">A61</font></i></a>, <a class="txt" href="#E7:110_2_2_2_1_1_2"><i><font color="Green" title="E100">A62</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:110_2_2_2_1_1_2_1_1"/><i><font color="Green" title="E101">A63</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_2_2"><i><font color="Green" title="E93">A52</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E3:110_2_2_2_1_1_2_1_1"/>
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q'</font>
 <b>by </b><i><a class="txt" href="#E6:110_2_2_2_1_1_2"><i><font color="Green" title="E99">A61</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c25">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_2_2_1_1_2_1_1"/>
(  not <font color="Maroon" title="c26">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">CRtfsm1</font> or <font color="Maroon" title="c26">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T78" target="_self" title="FSM_1:th.78">Th78</a></i>;<br/><b>hence </b><a NAME="E6:110_2_2_2_1_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2_2_1_1"><i><font color="Green" title="E95">A54</font></i></a>, <a class="txt" href="#E6:110_2_2_2_1_1_2"><i><font color="Green" title="E99">A61</font></i></a>, <a class="txt" href="#E1:110_2_2_2_1_1_2_1_1"><i><font color="Green" title="E101">A63</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:110_2_2_2_1_1_2_1_2"/><i><font color="Green" title="E101">A64</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:110_2_2_2_1_1_2_1_2"/><i><font color="Green" title="E102">A65</font></i>: 
<font color="Maroon" title="c25">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">Q'</font>
 <b>by </b><i><a class="txt" href="#E6:110_2_2_2_1_1_2"><i><font color="Green" title="E99">A61</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c27">y</font> = <font color="Maroon" title="c25">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E2:110_2_2_2_1_1_2_1_2"><i><font color="Green" title="E102">A65</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>;<br/><a NAME="E5:110_2_2_2_1_1_2_1_2"/>
(  not <font color="Maroon" title="c26">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  not <font color="Maroon" title="c27">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or <font color="Maroon" title="c26">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">y</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self" title="FSM_1:th.79">Th79</a></i>;<br/><b>hence </b><a NAME="E6:110_2_2_2_1_1_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2"><i><font color="Green" title="E89">A38</font></i></a>, <a class="txt" href="#E1:110_2_2_2_1_1_2_1_2"><i><font color="Green" title="E101">A64</font></i></a>, <a class="txt" href="#E2:110_2_2_2_1_1_2_1_2"><i><font color="Green" title="E102">A65</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:110_2_2_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:110_2_2_2_1_1"><i><font color="Green" title="E96">A55</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_2_2_2_1_2"><b>suppose </b></a><a NAME="E1:110_2_2_2_1_2"/><i><font color="Green" title="E95">A66</font></i>: 
<font color="Maroon" title="c21">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c22">q</font> = <font color="Maroon" title="c17">q</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E12:110"><i><font color="Green" title="E87">A7</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c23">x</font> = <font color="Maroon" title="c21">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c19">Q</font> <b>by </b><i><a class="txt" href="#E6:110_2_2_2"><i><font color="Green" title="E93">A52</font></i></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></i>;<br/><a NAME="E4:110_2_2_2_1_2"/>
(  not <font color="Maroon" title="c22">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or  not <font color="Maroon" title="c23">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">CRtfsm2</font> or <font color="Maroon" title="c22">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">x</font> )
 <b>by </b><i><a class="txt" href="#E1:110"><i><font color="Green" title="E81">A1</font></i></a>, <a class="ref" href="fsm_1.html#T79" target="_self" title="FSM_1:th.79">Th79</a></i>;<br/><b>hence </b><a NAME="E5:110_2_2_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:110_2_2"><i><font color="Green" title="E89">A38</font></i></a>, <a class="txt" href="#E6:110_2_2_2"><i><font color="Green" title="E93">A52</font></i></a>, <a class="txt" href="#E1:110_2_2_2_1_2"><i><font color="Green" title="E95">A66</font></i></a>, <a class="ref" href="zfmisc_1.html#T64" title="ZFMISC_1:th.64">ZFMISC_1:64</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:110_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E2:110_2_2"><i><font color="Green" title="E90">A39</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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