let n be Element of NAT ; :: thesis: for R being good Ring
for a being Data-Location of R
for loc being Instruction-Location of SCM R
for s1, s2 being State of st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
let R be good Ring; :: thesis: for a being Data-Location of R
for loc being Instruction-Location of SCM R
for s1, s2 being State of st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
let a be Data-Location of R; :: thesis: for loc being Instruction-Location of SCM R
for s1, s2 being State of st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
let loc be Instruction-Location of SCM R; :: thesis: for s1, s2 being State of st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
let s1, s2 be State of ; :: thesis: ( not R is trivial implies for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R ) )
assume A1: not R is trivial ; :: thesis: for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) holds
( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
set Cs2i1 = Computation s2,(n + 1);
set Cs1i1 = Computation s1,(n + 1);
set I = CurInstr (Computation s1,n);
let p be non NAT -defined autonomic FinPartState of ; :: thesis: ( p c= s1 & p c= s2 & CurInstr (Computation s1,n) = a =0_goto loc & loc <> Next (IC (Computation s1,n)) implies ( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R ) )
assume A2: ( p c= s1 & p c= s2 ) ; :: thesis: ( not CurInstr (Computation s1,n) = a =0_goto loc or not loc <> Next (IC (Computation s1,n)) or ( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R ) )
A3: CurInstr (Computation s1,n) = CurInstr (Computation s2,n) by A1, A2, Th28;
set Cs2i = Computation s2,n;
set Cs1i = Computation s1,n;
A4: Computation s1,(n + 1) = Following (Computation s1,n) by AMI_1:14
.= Exec (CurInstr (Computation s1,n)),(Computation s1,n) ;
A5: Computation s2,(n + 1) = Following (Computation s2,n) by AMI_1:14
.= Exec (CurInstr (Computation s2,n)),(Computation s2,n) ;
A6: ( ((Computation s1,(n + 1)) | (dom p)) . (IC (SCM R)) = (Computation s1,(n + 1)) . (IC (SCM R)) & ((Computation s2,(n + 1)) | (dom p)) . (IC (SCM R)) = (Computation s2,(n + 1)) . (IC (SCM R)) ) by A1, Th25, FUNCT_1:72;
assume that
A7: CurInstr (Computation s1,n) = a =0_goto loc and
A8: loc <> Next (IC (Computation s1,n)) ; :: thesis: ( (Computation s1,n) . a = 0. R iff (Computation s2,n) . a = 0. R )
A9: IC (Computation s1,n) = IC (Computation s2,n) by A1, A2, Th28;

hereby :: thesis: ( (Computation s2,n) . a = 0. R implies (Computation s1,n) . a = 0. R )

assume ( (Computation s1,n) . a = 0. R & (Computation s2,n) . a <> 0. R ) ; :: thesis: contradiction
then ( (Computation s1,(n + 1)) . (IC (SCM R)) = loc & (Computation s2,(n + 1)) . (IC (SCM R)) = Next (IC (Computation s2,n)) ) by A3, A4, A5, A7, SCMRING2:18;
hence contradiction by A2, A9, A6, A8, AMI_1:def 25; :: thesis: verum

end;

assume that
A10: (Computation s2,n) . a = 0. R and
A11: (Computation s1,n) . a <> 0. R ; :: thesis: contradiction
A12: (Computation s1,(n + 1)) . (IC (SCM R)) = Next (IC (Computation s1,n)) by A4, A7, A11, SCMRING2:18;
(Computation s2,(n + 1)) . (IC (SCM R)) = loc by A3, A5, A7, A10, SCMRING2:18;
hence contradiction by A2, A6, A8, A12, AMI_1:def 25; :: thesis: verum