let n be Nat; :: thesis: for I being Program of
for s1, s2 being State of SCMPDS
for P1, P2 being Instruction-Sequence of SCMPDS st s1 = s2 & I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s2,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s1,m) = Comput (P2,s2,m)

let I be Program of ; :: thesis: for s1, s2 being State of SCMPDS
for P1, P2 being Instruction-Sequence of SCMPDS st s1 = s2 & I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s2,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s1,m) = Comput (P2,s2,m)

let s1, s2 be State of SCMPDS; :: thesis: for P1, P2 being Instruction-Sequence of SCMPDS st s1 = s2 & I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s2,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s1,m) = Comput (P2,s2,m)

let P1, P2 be Instruction-Sequence of SCMPDS; :: thesis: ( s1 = s2 & I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s2,m)) in dom I ) implies for m being Nat st m <= n holds
Comput (P1,s1,m) = Comput (P2,s2,m) )

assume that
A1: s1 = s2 and
A2: I c= P1 and
A3: I c= P2 and
A4: for m being Nat st m < n holds
IC (Comput (P2,s2,m)) in dom I ; :: thesis: for m being Nat st m <= n holds
Comput (P1,s1,m) = Comput (P2,s2,m)

defpred S1[ Nat] means ( \$1 <= n implies Comput (P1,s1,\$1) = Comput (P2,s2,\$1) );
A5: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; :: thesis: ( S1[m] implies S1[m + 1] )
assume A6: ( m <= n implies Comput (P1,s1,m) = Comput (P2,s2,m) ) ; :: thesis: S1[m + 1]
A7: Comput (P2,s2,(m + 1)) = Following (P2,(Comput (P2,s2,m))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s2,m)))),(Comput (P2,s2,m))) ;
A8: Comput (P1,s1,(m + 1)) = Following (P1,(Comput (P1,s1,m))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s1,m)))),(Comput (P1,s1,m))) ;
assume A9: m + 1 <= n ; :: thesis: Comput (P1,s1,(m + 1)) = Comput (P2,s2,(m + 1))
A10: IC (Comput (P2,s2,m)) in dom I by ;
CurInstr (P1,(Comput (P1,s1,m))) = P1 . (IC (Comput (P1,s1,m))) by PBOOLE:143
.= I . (IC (Comput (P1,s1,m))) by
.= P2 . (IC (Comput (P2,s2,m))) by
.= CurInstr (P2,(Comput (P2,s2,m))) by PBOOLE:143 ;
hence Comput (P1,s1,(m + 1)) = Comput (P2,s2,(m + 1)) by A6, A8, A7, A9, NAT_1:13; :: thesis: verum
end;
Comput (P1,s1,0) = Comput (P2,s2,0) by A1;
then A11: S1[ 0 ] ;
thus for m being Nat holds S1[m] from NAT_1:sch 2(A11, A5); :: thesis: verum