let p be autonomic non programmed FinPartState of SCM+FSA ; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Int-Location
for loc being Instruction-Location of SCM+FSA st CurInstr (Computation s1,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 SCM+FSA ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da being Int-Location
for loc being Instruction-Location of SCM+FSA st CurInstr (Computation s1,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 Int-Location
for loc being Instruction-Location of SCM+FSA st CurInstr (Computation s1,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 Int-Location
for loc being Instruction-Location of SCM+FSA st CurInstr (Computation s1,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 Int-Location ; :: thesis: for loc being Instruction-Location of SCM+FSA st CurInstr (Computation s1,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+FSA ; :: thesis: ( CurInstr (Computation s1,i) = da =0_goto loc & loc <> Next (IC (Computation s1,i)) implies ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) )
set I = CurInstr (Computation s1,i);
set Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
A2: CurInstr (Computation s1,i) = CurInstr (Computation s2,i) by A1, Th18;
set Cs1i1 = Computation s1,(i + 1);
set Cs2i1 = Computation s2,(i + 1);
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+FSA ) = (Computation s1,(i + 1)) . (IC SCM+FSA ) & ((Computation s2,(i + 1)) | (dom p)) . (IC SCM+FSA ) = (Computation s2,(i + 1)) . (IC SCM+FSA ) ) by Th15, FUNCT_1:72;
A6: (Computation s1,(i + 1)) | (dom p) = (Computation s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
assume A7: ( CurInstr (Computation s1,i) = da =0_goto loc & loc <> Next (IC (Computation s1,i)) ) ; :: thesis: ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 )

A8: now

assume ( (Computation s1,i) . da = 0 & (Computation s2,i) . da <> 0 ) ; :: thesis: contradiction
then ( (Computation s1,(i + 1)) . (IC SCM+FSA ) = loc & (Computation s2,(i + 1)) . (IC SCM+FSA ) = Next (IC (Computation s2,i)) ) by A2, A3, A4, A7, SCMFSA_2:96;
hence contradiction by A1, A5, A6, A7, Th18; :: thesis: verum

end;

now

assume ( (Computation s2,i) . da = 0 & (Computation s1,i) . da <> 0 ) ; :: thesis: contradiction
then ( (Computation s2,(i + 1)) . (IC SCM+FSA ) = loc & (Computation s1,(i + 1)) . (IC SCM+FSA ) = Next (IC (Computation s1,i)) ) by A2, A3, A4, A7, SCMFSA_2:96;
hence contradiction by A1, A5, A7, AMI_1:def 25; :: thesis: verum

end;

hence ( (Computation s1,i) . da = 0 iff (Computation s2,i) . da = 0 ) by A8; :: thesis: verum