let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Data-Location
for loc being Instruction-Location of SCM
for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
let s1, s2 be State of ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da being Data-Location
for loc being Instruction-Location of SCM
for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da being Data-Location
for loc being Instruction-Location of SCM
for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
let i be Element of NAT ; :: thesis: for da being Data-Location
for loc being Instruction-Location of SCM
for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
let da be Data-Location ; :: thesis: for loc being Instruction-Location of SCM
for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
let loc be Instruction-Location of SCM ; :: thesis: for I being Instruction of st I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) holds
( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
let I be Instruction of ; :: thesis: ( I = CurInstr (Computation s1,i) & I = da =0_goto loc & loc <> Next (IC (Computation s1,i)) implies ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) )
assume A2: I = CurInstr (Computation s1,i) ; :: thesis: ( not I = da =0_goto loc or not loc <> Next (IC (Computation s1,i)) or ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) )
set Cs2i1 = Computation s2,(i + 1);
set Cs1i1 = Computation s1,(i + 1);
set Cs2i = Computation s2,i;
set Cs1i = Computation s1,i;
A3: Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
A4: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
A5: ( ((Computation s1,(i + 1)) | (dom p)) . (IC SCM ) = (Computation s1,(i + 1)) . (IC SCM ) & ((Computation s2,(i + 1)) | (dom p)) . (IC SCM ) = (Computation s2,(i + 1)) . (IC SCM ) ) by Th84, FUNCT_1:72;
assume that
A6: I = da =0_goto loc and
A7: loc <> Next (IC (Computation s1,i)) ; :: thesis: ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )
A8: I = CurInstr (Computation s2,i) by A1, A2, Th87;

A9: now

assume ( (Computation s2,i) . da = 0 & (Computation s1,i) . da <> 0 ) ; :: thesis: contradiction
then ( (Computation s2,(i + 1)) . (IC SCM ) = loc & (Computation s1,(i + 1)) . (IC SCM ) = Next (IC (Computation s1,i)) ) by A2, A8, A3, A4, A6, AMI_3:14;
hence contradiction by A1, A5, A7, AMI_1:def 25; :: thesis: verum

end;

A10: (Computation s1,(i + 1)) | (dom p) = (Computation s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;

now

assume ( (Computation s1,i) . da = 0 & (Computation s2,i) . da <> 0 ) ; :: thesis: contradiction
then ( (Computation s1,(i + 1)) . (IC SCM ) = loc & (Computation s2,(i + 1)) . (IC SCM ) = Next (IC (Computation s2,i)) ) by A2, A8, A3, A4, A6, AMI_3:14;
hence contradiction by A1, A2, A5, A10, A7, Th87; :: thesis: verum

end;

hence ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) by A9; :: thesis: verum