let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM+FSA st NPP p c= s1 & NPP p c= s2 holds
for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db)
let s1, s2 be State of SCM+FSA; :: thesis: ( NPP p c= s1 & NPP p c= s2 implies for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) )
assume B1: ( NPP p c= s1 & NPP p c= s2 ) ; :: thesis: for P1, P2 being the Instructions of SCM+FSA -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db)
let P1, P2 be the Instructions of SCM+FSA -valued ManySortedSet of NAT ; :: thesis: ( ProgramPart p c= P1 & ProgramPart p c= P2 implies for i being Element of NAT
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) )
assume A2: ( ProgramPart p c= P1 & ProgramPart p c= P2 ) ; :: thesis: for i being Element of NAT
for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db)
let i be Element of NAT ; :: thesis: for da, db being Int-Location st CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db holds
((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db)
let da, db be Int-Location ; :: thesis: ( CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) & da in dom p & da <> db implies ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) )
set I = CurInstr (P1,(Comput (P1,s1,i)));
set Cs1i = Comput (P1,s1,i);
set Cs2i = Comput (P2,s2,i);
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs2i1 = Comput (P2,s2,(i + 1));
A3: Comput (P2,s2,(i + 1)) = Following (P2,(Comput (P2,s2,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P2,(Comput (P2,s2,i)))),(Comput (P2,s2,i))) ;
A4: ( da in dom (NPP p) implies ( ((Comput (P1,s1,(i + 1))) | (dom (NPP p))) . da = (Comput (P1,s1,(i + 1))) . da & ((Comput (P2,s2,(i + 1))) | (dom (NPP p))) . da = (Comput (P2,s2,(i + 1))) . da ) ) by FUNCT_1:72;
assume that
A5: CurInstr (P1,(Comput (P1,s1,i))) = Divide (da,db) and
A6: da in dom p and
A7: da <> db and
A8: ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) <> ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) ; :: thesis: contradiction
A9: da in dom (NPP p) by A6, Lm1;
Comput (P1,s1,(i + 1)) = Following (P1,(Comput (P1,s1,i))) by EXTPRO_1:4
.= Exec ((CurInstr (P1,(Comput (P1,s1,i)))),(Comput (P1,s1,i))) ;
then A10: (Comput (P1,s1,(i + 1))) . da = ((Comput (P1,s1,i)) . da) div ((Comput (P1,s1,i)) . db) by A5, A7, SCMFSA_2:93;
CurInstr (P1,(Comput (P1,s1,i))) = CurInstr (P2,(Comput (P2,s2,i))) by B1, AMISTD_5:7, A2;
then (Comput (P2,s2,(i + 1))) . da = ((Comput (P2,s2,i)) . da) div ((Comput (P2,s2,i)) . db) by A3, A5, A7, SCMFSA_2:93;
hence contradiction by B1, A4, A8, A10, EXTPRO_1:def 9, A2, A9, B1; :: thesis: verum