let n be Element of NAT ; :: thesis: for R being good Ring
for a, b being Data-Location of R
for s1, s2 being State of (SCM R) st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p holds
((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b)
let R be good Ring; :: thesis: for a, b being Data-Location of R
for s1, s2 being State of (SCM R) st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p holds
((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b)
let a, b be Data-Location of R; :: thesis: for s1, s2 being State of (SCM R) st not R is trivial holds
for p being non NAT -defined autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p holds
((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b)
let s1, s2 be State of (SCM R); :: thesis: ( not R is trivial implies for p being non NAT -defined autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p holds
((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b) )
assume A1: not R is trivial ; :: thesis: for p being non NAT -defined autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p holds
((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b)
let p be non NAT -defined autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & p c= s2 & CurInstr (Computation s1,n) = MultBy a,b & a in dom p implies ((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b) )
assume A2: ( p c= s1 & p c= s2 ) ; :: thesis: ( not CurInstr (Computation s1,n) = MultBy a,b or not a in dom p or ((Computation s1,n) . a) * ((Computation s1,n) . b) = ((Computation s2,n) . a) * ((Computation s2,n) . b) )
set I = CurInstr (Computation s1,n);
set Cs1i = Computation s1,n;
set Cs2i = Computation s2,n;
A3: CurInstr (Computation s1,n) = CurInstr (Computation s2,n) by A1, A2, Th41;
set Cs1i1 = Computation s1,(n + 1);
set Cs2i1 = Computation s2,(n + 1);
A4: Computation s1,(n + 1) = Following (Computation s1,n) by AMI_1:14
.= Exec (CurInstr (Computation s1,n)),(Computation s1,n) ;
A5: Computation s2,(n + 1) = Following (Computation s2,n) by AMI_1:14
.= Exec (CurInstr (Computation s2,n)),(Computation s2,n) ;
A6: ( a in dom p implies ( ((Computation s1,(n + 1)) | (dom p)) . a = (Computation s1,(n + 1)) . a & ((Computation s2,(n + 1)) | (dom p)) . a = (Computation s2,(n + 1)) . a ) ) by FUNCT_1:72;
assume A7: ( CurInstr (Computation s1,n) = MultBy a,b & a in dom p & ((Computation s1,n) . a) * ((Computation s1,n) . b) <> ((Computation s2,n) . a) * ((Computation s2,n) . b) ) ; :: thesis: contradiction
then ( (Computation s1,(n + 1)) . a = ((Computation s1,n) . a) * ((Computation s1,n) . b) & (Computation s2,(n + 1)) . a = ((Computation s2,n) . a) * ((Computation s2,n) . b) ) by A3, A4, A5, SCMRING2:16;
hence contradiction by A2, A6, A7, AMI_1:def 25; :: thesis: verum