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, EXTPRO_1:def 9;
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 EXTPRO_1:4
.= 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 EXTPRO_1:4
.= 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