let p be non NAT -defined autonomic FinPartState of ; :: 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 Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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+FSA ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da being Int-Location
for loc being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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 Int-Location
for loc being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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 Int-Location
for loc being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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 Int-Location ; :: thesis: for loc being Element of NAT st CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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: ( CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,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 ) )
set Cs1i1 = Comput (ProgramPart s1),s1,(i + 1);
set Cs2i1 = Comput (ProgramPart s2),s2,(i + 1);
A2: (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;
set I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i);
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,i) by AMI_1:144;
A3: 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 ;
A4: ( ((Comput (ProgramPart s1),s1,(i + 1)) | (dom p)) . (IC SCM+FSA ) = (Comput (ProgramPart s1),s1,(i + 1)) . (IC SCM+FSA ) & ((Comput (ProgramPart s2),s2,(i + 1)) | (dom p)) . (IC SCM+FSA ) = (Comput (ProgramPart s2),s2,(i + 1)) . (IC SCM+FSA ) ) by Th15, FUNCT_1:72;
T: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,i) by AMI_1:144;
A5: 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 T ;
assume that
A6: CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = da >0_goto loc and
A7: loc <> succ (IC (Comput (ProgramPart s1),s1,i)) ; :: thesis: ( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 )
A8: CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) by A1, Th18;

A9: now

assume that
A10: (Comput (ProgramPart s2),s2,i) . da > 0 and
A11: (Comput (ProgramPart s1),s1,i) . da <= 0 ; :: thesis: contradiction
(Comput (ProgramPart s2),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 (Comput (ProgramPart s1),s1,i) = IC (Comput (ProgramPart s2),s2,i) by A1, Th18;

now

assume that
A13: (Comput (ProgramPart s1),s1,i) . da > 0 and
A14: (Comput (ProgramPart s2),s2,i) . da <= 0 ; :: thesis: contradiction
(Comput (ProgramPart s1),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 ( (Comput (ProgramPart s1),s1,i) . da > 0 iff (Comput (ProgramPart s2),s2,i) . da > 0 ) by A9; :: thesis: verum