let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
let s1, s2 be State of SCM ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),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 Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
let i be Element of NAT ; :: thesis: for da being Data-Location
for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
let da be Data-Location ; :: thesis: for loc being Element of NAT
for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
let loc be Element of NAT ; :: thesis: for I being Instruction of SCM st I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) holds
( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
let I be Instruction of SCM ; :: thesis: ( I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) & I = da >0_goto loc & loc <> succ (IC (Comput (ProgramPart s1),s1,i)) implies ( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 ) )
assume A2: I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) ; :: thesis: ( not I = da >0_goto loc or not loc <> succ (IC (Comput (ProgramPart s1),s1,i)) or ( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 ) )
set Cs2i1 = Comput (ProgramPart s2),s2,(i + 1);
set Cs1i1 = Comput (ProgramPart s1),s1,(i + 1);
A3: (Comput (ProgramPart s1),s1,(i + 1)) | (dom p) = (Comput (ProgramPart s2),s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
set Cs2i = Comput (ProgramPart s2),s2,i;
set Cs1i = Comput (ProgramPart s1),s1,i;
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,i) by AMI_1:123;
A4: Comput (ProgramPart s1),s1,(i + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) by T ;
S: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,i) by AMI_1:123;
A5: ( ((Comput (ProgramPart s1),s1,(i + 1)) | (dom p)) . (IC SCM ) = (Comput (ProgramPart s1),s1,(i + 1)) . (IC SCM ) & ((Comput (ProgramPart s2),s2,(i + 1)) | (dom p)) . (IC SCM ) = (Comput (ProgramPart s2),s2,(i + 1)) . (IC SCM ) ) by Th84, FUNCT_1:72;
A6: Comput (ProgramPart s2),s2,(i + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) by S ;
assume that
A7: I = da >0_goto loc and
A8: loc <> succ (IC (Comput (ProgramPart s1),s1,i)) ; :: thesis: ( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
A9: I = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) by A1, A2, Th87;
TX: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,i) by AMI_1:123;
TY: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,i) by AMI_1:123;

A10: now

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

end;

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

now

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

end;

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