let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM+FSA st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Int-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db & da in dom p holds
((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db)
let s1, s2 be State of SCM+FSA ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da, db being Int-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db & da in dom p holds
((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da, db being Int-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db & da in dom p holds
((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db)
let i be Element of NAT ; :: thesis: for da, db being Int-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db & da in dom p holds
((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db)
let da, db be Int-Location ; :: thesis: ( CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db & da in dom p implies ((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db) )
set I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i);
set Cs1i = Comput (ProgramPart s1),s1,i;
set Cs2i = Comput (ProgramPart s2),s2,i;
set Cs1i1 = Comput (ProgramPart s1),s1,(i + 1);
set Cs2i1 = Comput (ProgramPart s2),s2,(i + 1);
A2: Comput (ProgramPart s2),s2,(i + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) ;
A3: ( da in dom p implies ( ((Comput (ProgramPart s1),s1,(i + 1)) | (dom p)) . da = (Comput (ProgramPart s1),s1,(i + 1)) . da & ((Comput (ProgramPart s2),s2,(i + 1)) | (dom p)) . da = (Comput (ProgramPart s2),s2,(i + 1)) . da ) ) by FUNCT_1:72;
assume that
A4: CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = MultBy da,db and
A5: ( da in dom p & ((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) <> ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db) ) ; :: thesis: contradiction
Comput (ProgramPart s1),s1,(i + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) ;
then A6: (Comput (ProgramPart s1),s1,(i + 1)) . da = ((Comput (ProgramPart s1),s1,i) . da) * ((Comput (ProgramPart s1),s1,i) . db) by A4, SCMFSA_2:92;
CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = CurInstr (ProgramPart s2),(Comput (ProgramPart s2),s2,i) by A1, Th18;
then (Comput (ProgramPart s2),s2,(i + 1)) . da = ((Comput (ProgramPart s2),s2,i) . da) * ((Comput (ProgramPart s2),s2,i) . db) by A2, A4, SCMFSA_2:92;
hence contradiction by A1, A3, A5, A6, AMI_1:def 25; :: thesis: verum