let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st NPP p c= s1 & NPP p c= s2 holds
for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
let s1, s2 be State of SCM; :: thesis: ( NPP p c= s1 & NPP p c= s2 implies for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) )
assume B1: ( NPP p c= s1 & NPP p c= s2 ) ; :: thesis: for P1, P2 being the Instructions of SCM -valued ManySortedSet of NAT st ProgramPart p c= P1 & ProgramPart p c= P2 holds
for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
let P1, P2 be the Instructions of SCM -valued ManySortedSet of NAT ; :: thesis: ( ProgramPart p c= P1 & ProgramPart p c= P2 implies for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((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 Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
let i be Element of NAT ; :: thesis: for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
let da, db be Data-Location ; :: thesis: for I being Instruction of SCM st I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p holds
((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db)
let I be Instruction of SCM; :: thesis: ( I = CurInstr (P1,(Comput (P1,s1,i))) & I = AddTo (da,db) & da in dom p implies ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) )
assume A3: I = CurInstr (P1,(Comput (P1,s1,i))) ; :: thesis: ( not I = AddTo (da,db) or not da in dom p or ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) )
set Cs2i1 = Comput (P2,s2,(i + 1));
set Cs2i = Comput (P2,s2,i);
A4: 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))) ;
assume that
A5: I = AddTo (da,db) and
A6: ( da in dom p & ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) <> ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) ) ; :: thesis: contradiction
A7: da in dom (NPP p) by A6, Lm2;
I = CurInstr (P2,(Comput (P2,s2,i))) by A3, A2, B1, AMISTD_5:7;
then A8: (Comput (P2,s2,(i + 1))) . da = ((Comput (P2,s2,i)) . da) + ((Comput (P2,s2,i)) . db) by A4, A5, AMI_3:9;
set Cs1i1 = Comput (P1,s1,(i + 1));
set Cs1i = Comput (P1,s1,i);
A9: ( 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;
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 (Comput (P1,s1,(i + 1))) . da = ((Comput (P1,s1,i)) . da) + ((Comput (P1,s1,i)) . db) by A3, A5, AMI_3:9;
hence contradiction by A9, A6, A8, A2, A7, B1, EXTPRO_1:def 9; :: thesis: verum