let s1, s2 be State of SCM+FSA; :: thesis: for p1, p2 being Instruction-Sequence of SCM+FSA
for I being really-closed InitHalting Program of SCM+FSA st Initialize (() .--> 1) c= s1 & Initialize (() .--> 1) c= s2 & I c= p1 & I c= p2 & s1 = s2 holds
for k being Nat holds
( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )

let p1, p2 be Instruction-Sequence of SCM+FSA; :: thesis: for I being really-closed InitHalting Program of SCM+FSA st Initialize (() .--> 1) c= s1 & Initialize (() .--> 1) c= s2 & I c= p1 & I c= p2 & s1 = s2 holds
for k being Nat holds
( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )

let I be really-closed InitHalting Program of SCM+FSA; :: thesis: ( Initialize (() .--> 1) c= s1 & Initialize (() .--> 1) c= s2 & I c= p1 & I c= p2 & s1 = s2 implies for k being Nat holds
( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) ) )

assume that
A1: Initialize (() .--> 1) c= s1 and
A2: Initialize (() .--> 1) c= s2 and
A3: I c= p1 and
A4: I c= p2 and
A5: s1 = s2 ; :: thesis: for k being Nat holds
( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )

let k be Nat; :: thesis: ( Comput (p1,s1,k) = Comput (p2,s2,k) & CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k))) )
IC s1 = 0 by ;
then IC s1 in dom I by AFINSQ_1:65;
then A6: IC (Comput (p1,s1,k)) in dom I by ;
IC s2 = 0 by ;
then A7: IC s2 in dom I by AFINSQ_1:65;
then A8: IC (Comput (p2,s2,k)) in dom I by ;
for m being Nat st m < k holds
IC (Comput (p2,s2,m)) in dom I by ;
hence A9: Comput (p1,s1,k) = Comput (p2,s2,k) by ; :: thesis: CurInstr (p1,(Comput (p1,s1,k))) = CurInstr (p2,(Comput (p2,s2,k)))
thus CurInstr (p2,(Comput (p2,s2,k))) = p2 . (IC (Comput (p2,s2,k))) by PBOOLE:143
.= I . (IC (Comput (p2,s2,k))) by
.= p1 . (IC (Comput (p1,s1,k))) by
.= CurInstr (p1,(Comput (p1,s1,k))) by PBOOLE:143 ; :: thesis: verum