let n be Nat; :: thesis: for R being non trivial Ring
for a, b being Data-Location of R
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM b1) -valued finite non halt-free Function
for p being non empty b8 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

let R be non trivial Ring; :: thesis: for a, b being Data-Location of R
for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b7 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

let a, b be Data-Location of R; :: thesis: for s1, s2 being State of (SCM R)
for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b5 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

let s1, s2 be State of (SCM R); :: thesis: for P1, P2 being Instruction-Sequence of (SCM R)
for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b3 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

let P1, P2 be Instruction-Sequence of (SCM R); :: thesis: for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function
for p being non empty b1 -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

set Cs2i1 = Comput (P2,s2,(n + 1));
set Cs1i1 = Comput (P1,s1,(n + 1));
set Cs2i = Comput (P2,s2,n);
set Cs1i = Comput (P1,s1,n);
set I = CurInstr (P1,(Comput (P1,s1,n)));
let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p holds
((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b)

let p be non empty q -autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) & a in dom p implies ((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b) )
assume that
A1: ( p c= s1 & p c= s2 ) and
A2: ( q c= P1 & q c= P2 ) ; :: thesis: ( not CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) or not a in dom p or ((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b) )
A3: ( a in dom p implies ( ((Comput (P1,s1,(n + 1))) | (dom p)) . a = (Comput (P1,s1,(n + 1))) . a & ((Comput (P2,s2,(n + 1))) | (dom p)) . a = (Comput (P2,s2,(n + 1))) . a ) ) by FUNCT_1:49;
A4: Comput (P2,s2,(n + 1)) = Following (P2,(Comput (P2,s2,n))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n))) ;
assume that
A5: CurInstr (P1,(Comput (P1,s1,n))) = AddTo (a,b) and
A6: ( a in dom p & ((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) <> ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b) ) ; :: thesis: contradiction
Comput (P1,s1,(n + 1)) = Following (P1,(Comput (P1,s1,n))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n))) ;
then A7: (Comput (P1,s1,(n + 1))) . a = ((Comput (P1,s1,n)) . a) + ((Comput (P1,s1,n)) . b) by ;
CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by ;
then (Comput (P2,s2,(n + 1))) . a = ((Comput (P2,s2,n)) . a) + ((Comput (P2,s2,n)) . b) by ;
hence contradiction by A1, A3, A6, A7, A2, EXTPRO_1:def 10; :: thesis: verum