let p be non NAT -defined autonomic FinPartState of ; :: thesis: for s1, s2 being State of SCM st p c= s1 & p c= s2 holds
for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db holds
((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db)
let s1, s2 be State of SCM ; :: thesis: ( p c= s1 & p c= s2 implies for i being Element of NAT
for da, db being Data-Location
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db holds
((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((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 Data-Location
for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db holds
((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),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 (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db holds
((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db)
let da, db be Data-Location ; :: thesis: for I being Instruction of SCM st I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db holds
((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db)
let I be Instruction of SCM ; :: thesis: ( I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) & I = Divide da,db & da in dom p & da <> db implies ((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db) )
assume A2: I = CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) ; :: thesis: ( not I = Divide da,db or not da in dom p or not da <> db or ((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db) )
set Cs2i1 = Comput (ProgramPart s2),s2,(i + 1);
set Cs2i = Comput (ProgramPart s2),s2,i;
T: ProgramPart s2 = ProgramPart (Comput (ProgramPart s2),s2,i) by AMI_1:144;
A3: Comput (ProgramPart s2),s2,(i + 1) = Following (ProgramPart s2),(Comput (ProgramPart s2),s2,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) by T ;
assume that
A4: I = Divide da,db and
A5: da in dom p and
A6: da <> db and
A7: ((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) <> ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db) ; :: thesis: contradiction
I = CurInstr (ProgramPart (Comput (ProgramPart s2),s2,i)),(Comput (ProgramPart s2),s2,i) by A1, A2, Th87;
then A8: (Comput (ProgramPart s2),s2,(i + 1)) . da = ((Comput (ProgramPart s2),s2,i) . da) div ((Comput (ProgramPart s2),s2,i) . db) by A3, A4, A6, AMI_3:12;
set Cs1i1 = Comput (ProgramPart s1),s1,(i + 1);
set Cs1i = Comput (ProgramPart s1),s1,i;
T: ProgramPart s1 = ProgramPart (Comput (ProgramPart s1),s1,i) by AMI_1:144;
A9: ( 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;
Comput (ProgramPart s1),s1,(i + 1) = Following (ProgramPart s1),(Comput (ProgramPart s1),s1,i) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i)),(Comput (ProgramPart s1),s1,i) by T ;
then (Comput (ProgramPart s1),s1,(i + 1)) . da = ((Comput (ProgramPart s1),s1,i) . da) div ((Comput (ProgramPart s1),s1,i) . db) by A2, A4, A6, AMI_3:12;
hence contradiction by A1, A9, A5, A7, A8, AMI_1:def 25; :: thesis: verum