let p be autonomic non programmed FinPartState of SCM+FSA ; :: 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 (Computation s1,i) = Divide da,db & db in dom p & da <> db holds
((Computation s1,i) . da) mod ((Computation s1,i) . db) = ((Computation s2,i) . da) mod ((Computation 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 (Computation s1,i) = Divide da,db & db in dom p & da <> db holds
((Computation s1,i) . da) mod ((Computation s1,i) . db) = ((Computation s2,i) . da) mod ((Computation 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 (Computation s1,i) = Divide da,db & db in dom p & da <> db holds
((Computation s1,i) . da) mod ((Computation s1,i) . db) = ((Computation s2,i) . da) mod ((Computation s2,i) . db)
let i be Element of NAT ; :: thesis: for da, db being Int-Location st CurInstr (Computation s1,i) = Divide da,db & db in dom p & da <> db holds
((Computation s1,i) . da) mod ((Computation s1,i) . db) = ((Computation s2,i) . da) mod ((Computation s2,i) . db)
let da, db be Int-Location ; :: thesis: ( CurInstr (Computation s1,i) = Divide da,db & db in dom p & da <> db implies ((Computation s1,i) . da) mod ((Computation s1,i) . db) = ((Computation s2,i) . da) mod ((Computation s2,i) . db) )
set I = CurInstr (Computation s1,i);
set Cs1i = Computation s1,i;
set Cs2i = Computation s2,i;
A2: CurInstr (Computation s1,i) = CurInstr (Computation s2,i) by A1, Th18;
set Cs1i1 = Computation s1,(i + 1);
set Cs2i1 = Computation s2,(i + 1);
A3: Computation s1,(i + 1) = Following (Computation s1,i) by AMI_1:14
.= Exec (CurInstr (Computation s1,i)),(Computation s1,i) ;
A4: Computation s2,(i + 1) = Following (Computation s2,i) by AMI_1:14
.= Exec (CurInstr (Computation s2,i)),(Computation s2,i) ;
assume A5: ( CurInstr (Computation s1,i) = Divide da,db & db in dom p & da <> db & ((Computation s1,i) . da) mod ((Computation s1,i) . db) <> ((Computation s2,i) . da) mod ((Computation s2,i) . db) ) ; :: thesis: contradiction
then A6: ( ((Computation s1,(i + 1)) | (dom p)) . db = (Computation s1,(i + 1)) . db & ((Computation s2,(i + 1)) | (dom p)) . db = (Computation s2,(i + 1)) . db ) by FUNCT_1:72;
( (Computation s1,(i + 1)) . db = ((Computation s1,i) . da) mod ((Computation s1,i) . db) & (Computation s2,(i + 1)) . db = ((Computation s2,i) . da) mod ((Computation s2,i) . db) ) by A2, A3, A4, A5, SCMFSA_2:93;
hence contradiction by A1, A5, A6, AMI_1:def 25; :: thesis: verum