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 b_{1}) -valued finite non halt-free Function

for p being non empty b_{8} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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 b_{7} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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 b_{5} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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 b_{3} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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 b_{1} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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))) = a := b & a in dom p implies (Comput (P1,s1,n)) . b = (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))) = a := b or not a in dom p or (Comput (P1,s1,n)) . b = (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))) = a := b and

A6: ( a in dom p & (Comput (P1,s1,n)) . b <> (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)) . b by A5, SCMRING2:11;

CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by A1, A2, AMISTD_5:7;

then (Comput (P2,s2,(n + 1))) . a = (Comput (P2,s2,n)) . b by A4, A5, SCMRING2:11;

hence contradiction by A1, A3, A6, A7, A2, EXTPRO_1:def 10; :: thesis: verum

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 b

for p being non empty b

(Comput (P1,s1,n)) . b = (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 b

(Comput (P1,s1,n)) . b = (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 b

(Comput (P1,s1,n)) . b = (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 b

(Comput (P1,s1,n)) . b = (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 b

(Comput (P1,s1,n)) . b = (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))) = a := b & a in dom p holds

(Comput (P1,s1,n)) . b = (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))) = a := b & a in dom p implies (Comput (P1,s1,n)) . b = (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))) = a := b or not a in dom p or (Comput (P1,s1,n)) . b = (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))) = a := b and

A6: ( a in dom p & (Comput (P1,s1,n)) . b <> (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)) . b by A5, SCMRING2:11;

CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by A1, A2, AMISTD_5:7;

then (Comput (P2,s2,(n + 1))) . a = (Comput (P2,s2,n)) . b by A4, A5, SCMRING2:11;

hence contradiction by A1, A3, A6, A7, A2, EXTPRO_1:def 10; :: thesis: verum