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 being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

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 being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

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 being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0 )

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 being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

let P1, P2 be Instruction-Sequence of SCM+FSA; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0 )

assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

let i be Nat; :: thesis: for da being Int-Location
for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

let da be Int-Location; :: thesis: for f being FinSeq-Location st CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p holds
for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

let f be FinSeq-Location ; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = f :=<0,...,0> da & f in dom p implies for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0 )

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 ;
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: ( f in dom p implies ( ((Comput (P1,s1,(i + 1))) | (dom p)) . f = (Comput (P1,s1,(i + 1))) . f & ((Comput (P2,s2,(i + 1))) | (dom p)) . f = (Comput (P2,s2,(i + 1))) . f ) ) 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))) = f :=<0,...,0> da and
A8: f in dom p ; :: thesis: for k1, k2 being Nat st k1 = |.((Comput (P1,s1,i)) . da).| & k2 = |.((Comput (P2,s2,i)) . da).| holds
k1 |-> 0 = k2 |-> 0

A9: ( ex k1 being Nat st
( k1 = |.((Comput (P1,s1,i)) . da).| & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i)))) . f = k1 |-> 0 ) & ex k2 being Nat st
( k2 = |.((Comput (P2,s2,i)) . da).| & (Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P2,s2,i)))) . f = k2 |-> 0 ) ) by ;
let i1, i2 be Nat; :: thesis: ( i1 = |.((Comput (P1,s1,i)) . da).| & i2 = |.((Comput (P2,s2,i)) . da).| implies i1 |-> 0 = i2 |-> 0 )
assume ( i1 = |.((Comput (P1,s1,i)) . da).| & i2 = |.((Comput (P2,s2,i)) . da).| & i1 |-> 0 <> i2 |-> 0 ) ; :: thesis: contradiction
hence contradiction by A1, A4, A6, A5, A3, A9, A8, A2, AMISTD_5:7; :: thesis: verum