let q be NAT -defined the InstructionsF 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 Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
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 Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
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 Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db )
assume A1: ( 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 Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db )
assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
let i be Nat; :: thesis: for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p holds
(Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db
let da, db be Int-Location; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = da := db & da in dom p implies (Comput (P1,s1,i)) . db = (Comput (P2,s2,i)) . db )
set I = CurInstr (P1,(Comput (P1,s1,i)));
set Cs1i = Comput (P1,s1,i);
set Cs2i = Comput (P2,s2,i);
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: 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))) ;
A4: ( 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;
assume that
A5: CurInstr (P1,(Comput (P1,s1,i))) = da := db and
A6: ( da in dom p & (Comput (P1,s1,i)) . db <> (Comput (P2,s2,i)) . db ) ; :: thesis: contradiction
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))) ;
then A7: (Comput (P1,s1,(i + 1))) . da = (Comput (P1,s1,i)) . db by A5, SCMFSA_2:63;
CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) by A1, A2, AMISTD_5:7;
then (Comput (P2,s2,(i + 1))) . da = (Comput (P2,s2,i)) . db by A3, A5, SCMFSA_2:63;
hence contradiction by A4, A6, A7, A2, A1, EXTPRO_1:def 10; :: thesis: verum