let q be NAT -defined the InstructionsF of SCM -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of SCM
for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let p be non empty q -autonomic FinPartState of SCM; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let s1, s2 be State of SCM; :: thesis: ( p c= s1 & p c= s2 implies for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 ) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for P1, P2 being Instruction-Sequence of SCM st q c= P1 & q c= P2 holds
for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let P1, P2 be Instruction-Sequence of SCM; :: thesis: ( q c= P1 & q c= P2 implies for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 ) )
assume A2: ( q c= P1 & q c= P2 ) ; :: thesis: for i being Nat
for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let i be Nat; :: thesis: for da being Data-Location
for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let da be Data-Location; :: thesis: for loc being Nat
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let loc be Nat; :: thesis: for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 holds
( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
let I be Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = da >0_goto loc & loc <> (IC (Comput (P1,s1,i))) + 1 implies ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 ) )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = da >0_goto loc or not loc <> (IC (Comput (P1,s1,i))) + 1 or ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 ) )
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs1i1 = Comput (P1,s1,(i + 1));
A4: (Comput (P1,s1,(i + 1))) | (dom p) = (Comput (P2,s2,(i + 1))) | (dom p) by A2, A1, EXTPRO_1:def 10;
set Cs2i = Comput (P2,s2,i);
set Cs1i = Comput (P1,s1,i);
A5: Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
IC in dom p by AMISTD_5:6;
then A6: ( ((Comput (P1,s1,(i + 1))) | (dom p)) . (IC ) = IC (Comput (P1,s1,(i + 1))) & ((Comput (P2,s2,(i + 1))) | (dom p)) . (IC ) = IC (Comput (P2,s2,(i + 1))) ) by FUNCT_1:49;
A7: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
assume that
A8: I = da >0_goto loc and
A9: loc <> (IC (Comput (P1,s1,i))) + 1 ; :: thesis: ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 )
A10: I = CurInstr (P2,(Comput (P2,s2,i))) by A3, A2, A1, AMISTD_5:7;

A11: now :: thesis: ( (Comput (P2,s2,i)) . da > 0 implies not (Comput (P1,s1,i)) . da <= 0 )

assume that
A12: (Comput (P2,s2,i)) . da > 0 and
A13: (Comput (P1,s1,i)) . da <= 0 ; :: thesis: contradiction
(Comput (P2,s2,(i + 1))) . (IC ) = loc by A10, A7, A8, A12, AMI_3:9;
hence contradiction by A3, A5, A6, A4, A8, A9, A13, AMI_3:9; :: thesis: verum

end;

A14: IC (Comput (P1,s1,i)) = IC (Comput (P2,s2,i)) by A2, A1, AMISTD_5:7;

now :: thesis: ( (Comput (P1,s1,i)) . da > 0 implies not (Comput (P2,s2,i)) . da <= 0 )

assume that
A15: (Comput (P1,s1,i)) . da > 0 and
A16: (Comput (P2,s2,i)) . da <= 0 ; :: thesis: contradiction
(Comput (P1,s1,(i + 1))) . (IC ) = loc by A3, A5, A8, A15, AMI_3:9;
hence contradiction by A14, A10, A7, A6, A4, A8, A9, A16, AMI_3:9; :: thesis: verum

end;

hence ( (Comput (P1,s1,i)) . da > 0 iff (Comput (P2,s2,i)) . da > 0 ) by A11; :: thesis: verum