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);
A3: (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;
A4: Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,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;
A6: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
assume that
A7: I = da >0_goto loc and
A8: loc <> Next (IC (Computation s1,i)) ; :: thesis: ( (Computation s1,i) . da > 0 iff (Computation s2,i) . da > 0 )
A9: I = CurInstr (Computation s2,i) by A1, A2, Th87;

A10: now

assume that
A11: (Computation s2,i) . da > 0 and
A12: (Computation s1,i) . da <= 0 ; :: thesis: contradiction
(Computation s2,(i + 1)) . (IC SCM ) = loc by A9, A6, A7, A11, AMI_3:15;
hence contradiction by A2, A4, A5, A3, A7, A8, A12, AMI_3:15; :: thesis: verum

end;

A13: IC (Computation s1,i) = IC (Computation s2,i) by A1, A2, Th87;

now

assume that
A14: (Computation s1,i) . da > 0 and
A15: (Computation s2,i) . da <= 0 ; :: thesis: contradiction
(Computation s1,(i + 1)) . (IC SCM ) = loc by A2, A4, A7, A14, AMI_3:15;
hence contradiction by A13, A9, A6, A5, A3, A7, A8, A15, AMI_3:15; :: thesis: verum

end;

hence ( (Computation s1,i) . da > 0 iff (Computation s2,i) . da > 0 ) by A10; :: thesis: verum