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 st p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p holds
(Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),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 st p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p holds
(Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),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 st p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p holds
(Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),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 st p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p holds
(Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),s2,n)) . b )
assume A1: not R is trivial ; :: thesis: for p being non NAT -defined autonomic FinPartState of st p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p holds
(Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),s2,n)) . b
set Cs2i1 = Comput ((ProgramPart s2),s2,(n + 1));
set Cs1i1 = Comput ((ProgramPart s1),s1,(n + 1));
set Cs2i = Comput ((ProgramPart s2),s2,n);
set Cs1i = Comput ((ProgramPart s1),s1,n);
set I = CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n)));
let p be non NAT -defined autonomic FinPartState of ; :: thesis: ( p c= s1 & p c= s2 & CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b & a in dom p implies (Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),s2,n)) . b )
assume A2: ( p c= s1 & p c= s2 ) ; :: thesis: ( not CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b or not a in dom p or (Comput ((ProgramPart s1),s1,n)) . b = (Comput ((ProgramPart s2),s2,n)) . b )
A3: ( a in dom p implies ( ((Comput ((ProgramPart s1),s1,(n + 1))) | (dom p)) . a = (Comput ((ProgramPart s1),s1,(n + 1))) . a & ((Comput ((ProgramPart s2),s2,(n + 1))) | (dom p)) . a = (Comput ((ProgramPart s2),s2,(n + 1))) . a ) ) by FUNCT_1:72;
A4: Comput ((ProgramPart s2),s2,(n + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,n))),(Comput ((ProgramPart s2),s2,n)))),(Comput ((ProgramPart s2),s2,n))) by AMI_1:123 ;
assume that
A5: CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = a := b and
A6: ( a in dom p & (Comput ((ProgramPart s1),s1,n)) . b <> (Comput ((ProgramPart s2),s2,n)) . b ) ; :: thesis: contradiction
Comput ((ProgramPart s1),s1,(n + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,n))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n)))),(Comput ((ProgramPart s1),s1,n))) by AMI_1:123 ;
then A7: (Comput ((ProgramPart s1),s1,(n + 1))) . a = (Comput ((ProgramPart s1),s1,n)) . b by A5, SCMRING2:13;
CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,n))),(Comput ((ProgramPart s1),s1,n))) = CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,n))),(Comput ((ProgramPart s2),s2,n))) by A1, A2, Th28;
then (Comput ((ProgramPart s2),s2,(n + 1))) . a = (Comput ((ProgramPart s2),s2,n)) . b by A4, A5, SCMRING2:13;
hence contradiction by A2, A3, A6, A7, EXTPRO_1:def 9; :: thesis: verum