let q be NAT -defined the Instructions of SCM+FSA -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of SCM+FSA
for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let p be non empty q -autonomic FinPartState of SCM+FSA; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let s1, s2 be State of SCM+FSA; :: thesis: ( p c= s1 & p c= s2 implies for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2 )
assume B1: ( p c= s1 & p c= s2 ) ; :: thesis: for P1, P2 being Instruction-Sequence of SCM+FSA st q c= P1 & q c= P2 holds
for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2 )
assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let i be Element of NAT ; :: thesis: for da, db being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let da, db be Int-Location ; :: thesis: for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
let f be FinSeq-Location ; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) & da in dom p implies for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2 )
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: (Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p) by B1, A2, EXTPRO_1:def 10;
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
set I = CurInstr (P1,(Comput (P1,s1,i)));
A4: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
A5: ( da in dom p implies ( ((Comput (P1,s1,(i + 1))) | (dom p)) . da = (Comput (P1,s1,(i + 1))) . da & ((Comput (P2,s2,(i + 1))) | (dom p)) . da = (Comput (P2,s2,(i + 1))) . da ) ) by FUNCT_1:49;
A6: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
assume that
A7: CurInstr (P1,(Comput (P1,s1,i))) = da := (f,db) and
A8: da in dom p ; :: thesis: for k1, k2 being Element of NAT st k1 = abs ((Comput (P1,s1,i)) . db) & k2 = abs ((Comput (P2,s2,i)) . db) holds
((Comput (P1,s1,i)) . f) /. k1 = ((Comput (P2,s2,i)) . f) /. k2
A9: ( ex k1 being Element of NAT st
( k1 = abs ((Comput (P1,s1,i)) . db) & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . da = ((Comput (P1,s1,i)) . f) /. k1 ) & ex k2 being Element of NAT st
( k2 = abs ((Comput (P2,s2,i)) . db) & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P2,s2,i)))) . da = ((Comput (P2,s2,i)) . f) /. k2 ) ) by A7, SCMFSA_2:72;
let i1, i2 be Element of NAT ; :: thesis: ( i1 = abs ((Comput (P1,s1,i)) . db) & i2 = abs ((Comput (P2,s2,i)) . db) implies ((Comput (P1,s1,i)) . f) /. i1 = ((Comput (P2,s2,i)) . f) /. i2 )
assume ( i1 = abs ((Comput (P1,s1,i)) . db) & i2 = abs ((Comput (P2,s2,i)) . db) & ((Comput (P1,s1,i)) . f) /. i1 <> ((Comput (P2,s2,i)) . f) /. i2 ) ; :: thesis: contradiction
hence contradiction by B1, A4, A6, A5, A3, A9, A8, A2, AMISTD_5:7; :: thesis: verum