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
for f being FinSeq-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da)
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
for f being FinSeq-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da) )
assume A1: ( p c= s1 & p c= s2 ) ; :: thesis: for i being Element of NAT
for da, db being Int-Location
for f being FinSeq-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da)
let i be Element of NAT ; :: thesis: for da, db being Int-Location
for f being FinSeq-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da)
let da, db be Int-Location ; :: thesis: for f being FinSeq-Location st CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p holds
for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da)
let f be FinSeq-Location ; :: thesis: ( CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da & f in dom p implies for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da) )
set Cs1i1 = Comput (ProgramPart s1),s1,(i + 1);
set Cs2i1 = Comput (ProgramPart s2),s2,(i + 1);
A2: (Comput (ProgramPart s1),s1,(i + 1)) | (dom p) = (Comput (ProgramPart s2),s2,(i + 1)) | (dom p) by A1, AMI_1:def 25;
set Cs2i = Comput (ProgramPart s2),s2,i;
set Cs1i = Comput (ProgramPart s1),s1,i;
set I = CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i);
A3: 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) ;
A4: ( f in dom p implies ( ((Comput (ProgramPart s1),s1,(i + 1)) | (dom p)) . f = (Comput (ProgramPart s1),s1,(i + 1)) . f & ((Comput (ProgramPart s2),s2,(i + 1)) | (dom p)) . f = (Comput (ProgramPart s2),s2,(i + 1)) . f ) ) by FUNCT_1:72;
A5: 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) ;
assume that
A6: CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i) = f,db := da and
A7: f in dom p ; :: thesis: for k1, k2 being Element of NAT st k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & k2 = abs ((Comput (ProgramPart s2),s2,i) . db) holds
((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da)
A8: ( ex k1 being Element of NAT st
( k1 = abs ((Comput (ProgramPart s1),s1,i) . db) & (Exec (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i)) . f = ((Comput (ProgramPart s1),s1,i) . f) +* k1,((Comput (ProgramPart s1),s1,i) . da) ) & ex k2 being Element of NAT st
( k2 = abs ((Comput (ProgramPart s2),s2,i) . db) & (Exec (CurInstr (ProgramPart s1),(Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s2),s2,i)) . f = ((Comput (ProgramPart s2),s2,i) . f) +* k2,((Comput (ProgramPart s2),s2,i) . da) ) ) by A6, SCMFSA_2:99;
let i1, i2 be Element of NAT ; :: thesis: ( i1 = abs ((Comput (ProgramPart s1),s1,i) . db) & i2 = abs ((Comput (ProgramPart s2),s2,i) . db) implies ((Comput (ProgramPart s1),s1,i) . f) +* i1,((Comput (ProgramPart s1),s1,i) . da) = ((Comput (ProgramPart s2),s2,i) . f) +* i2,((Comput (ProgramPart s2),s2,i) . da) )
assume ( i1 = abs ((Comput (ProgramPart s1),s1,i) . db) & i2 = abs ((Comput (ProgramPart s2),s2,i) . db) & ((Comput (ProgramPart s1),s1,i) . f) +* i1,((Comput (ProgramPart s1),s1,i) . da) <> ((Comput (ProgramPart s2),s2,i) . f) +* i2,((Comput (ProgramPart s2),s2,i) . da) ) ; :: thesis: contradiction
hence contradiction by A1, A3, A5, A4, A2, A7, A8, Th18; :: thesis: verum