let p be non NAT -defined autonomic FinPartState of SCM ; :: 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 Instruction-Location of SCM
for I being Instruction of SCM 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 SCM ; :: 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 SCM 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 SCM 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 SCM 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 SCM 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 SCM 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 SCM ; :: 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 Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
A3: IC (Computation s1,i) = IC (Computation s2,i) by A1, A2, Th87;
A4: I = CurInstr (Computation s2,i) by A1, A2, Th87;
set Cs1i1 = Computation s1,(i + 1);
set Cs2i1 = Computation s2,(i + 1);
A5: Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
A6: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
A7: ( ((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;
A8: (Computation s1,(i + 1)) | (dom p) = (Computation s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
assume A9: ( I = da >0_goto loc & loc <> Next (IC (Computation s1,i)) ) ; :: thesis: ( (Computation s1,i) . da > 0 iff (Computation s2,i) . da > 0 )

A10: now

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

end;

now

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

end;

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