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, EXTPRO_1:def 9;
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 EXTPRO_1:4
.= 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 EXTPRO_1:4
.= 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