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 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 ; :: 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 Cs1i1 = Computation s1,(i + 1);
set Cs2i1 = Computation s2,(i + 1);
A2: (Computation s1,(i + 1)) | (dom p) = (Computation s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
set Cs2i = Computation s2,i;
set Cs1i = Computation s1,i;
set I = CurInstr (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 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;
A5: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
assume that
A6: CurInstr (Computation s1,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: CurInstr (Computation s1,i) = CurInstr (Computation s2,i) by A1, Th18;

A9: now

assume that
A10: (Computation s2,i) . da > 0 and
A11: (Computation s1,i) . da <= 0 ; :: thesis: contradiction
(Computation s2,(i + 1)) . (IC SCM+FSA ) = loc by A8, A5, A6, A10, SCMFSA_2:97;
hence contradiction by A3, A4, A2, A6, A7, A11, SCMFSA_2:97; :: thesis: verum

end;

A12: IC (Computation s1,i) = IC (Computation s2,i) by A1, Th18;

now

assume that
A13: (Computation s1,i) . da > 0 and
A14: (Computation s2,i) . da <= 0 ; :: thesis: contradiction
(Computation s1,(i + 1)) . (IC SCM+FSA ) = loc by A3, A6, A13, SCMFSA_2:97;
hence contradiction by A12, A8, A5, A4, A2, A6, A7, A14, SCMFSA_2:97; :: thesis: verum

end;

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