let n be Nat; :: thesis: for R being non trivial Ring
for a being Data-Location of R
for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM b1) -valued finite non halt-free Function
for p being non empty b8 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let R be non trivial Ring; :: thesis: for a being Data-Location of R
for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b7 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let a be Data-Location of R; :: thesis: for loc being Nat
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b6 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let loc be Nat; :: thesis: for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b5 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let s1, s2 be State of (SCM R); :: thesis: for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b3 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let P1, P2 be Instruction-Sequence of (SCM R); :: thesis: for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b1 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

set Cs2i1 = Comput (P2,s2,(n + 1));
set Cs1i1 = Comput (P1,s1,(n + 1));
set I = CurInstr (P1,(Comput (P1,s1,n)));
let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds
( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let p be non empty q -autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 implies ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )
assume that
A1: ( p c= s1 & p c= s2 ) and
A2: ( q c= P1 & q c= P2 ) ; :: thesis: ( not CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc or not loc <> (IC (Comput (P1,s1,n))) + 1 or ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )
A3: CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by ;
set Cs2i = Comput (P2,s2,n);
set Cs1i = Comput (P1,s1,n);
A4: Comput (P1,s1,(n + 1)) = Following (P1,(Comput (P1,s1,n))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n))) ;
A5: Comput (P2,s2,(n + 1)) = Following (P2,(Comput (P2,s2,n))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n))) ;
IC in dom p by AMISTD_5:6;
then A6: ( ((Comput (P1,s1,(n + 1))) | (dom p)) . () = (Comput (P1,s1,(n + 1))) . () & ((Comput (P2,s2,(n + 1))) | (dom p)) . () = (Comput (P2,s2,(n + 1))) . () ) by FUNCT_1:49;
assume that
A7: CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc and
A8: loc <> (IC (Comput (P1,s1,n))) + 1 ; :: thesis: ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )
A9: IC (Comput (P1,s1,n)) = IC (Comput (P2,s2,n)) by ;
hereby :: thesis: ( (Comput (P2,s2,n)) . a = 0. R implies (Comput (P1,s1,n)) . a = 0. R )
assume ( (Comput (P1,s1,n)) . a = 0. R & (Comput (P2,s2,n)) . a <> 0. R ) ; :: thesis: contradiction
then ( (Comput (P1,s1,(n + 1))) . () = loc & (Comput (P2,s2,(n + 1))) . () = (IC (Comput (P2,s2,n))) + 1 ) by ;
hence contradiction by A1, A9, A6, A8, A2, EXTPRO_1:def 10; :: thesis: verum
end;
assume that
A10: (Comput (P2,s2,n)) . a = 0. R and
A11: (Comput (P1,s1,n)) . a <> 0. R ; :: thesis: contradiction
A12: (Comput (P1,s1,(n + 1))) . () = (IC (Comput (P1,s1,n))) + 1 by ;
(Comput (P2,s2,(n + 1))) . () = loc by ;
hence contradiction by A1, A6, A8, A12, A2, EXTPRO_1:def 10; :: thesis: verum