let N be with_zero set ; :: thesis: for n being Nat
for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N
for s being State of S
for I being Program of
for P1, P2 being Instruction-Sequence of S st I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)

let n be Nat; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated AMI-Struct over N
for s being State of S
for I being Program of
for P1, P2 being Instruction-Sequence of S st I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)

let S be non empty with_non-empty_values IC-Ins-separated AMI-Struct over N; :: thesis: for s being State of S
for I being Program of
for P1, P2 being Instruction-Sequence of S st I c= P1 & I c= P2 & ( for m being Nat st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)

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

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

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

assume A1: ( I c= P1 & I c= P2 ) ; :: thesis: ( ex m being Nat st
( m < n & not IC (Comput (P2,s,m)) in dom I ) or for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m) )

assume A2: for m being Nat st m < n holds
IC (Comput (P2,s,m)) in dom I ; :: thesis: for m being Nat st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)

defpred S1[ Nat] means ( \$1 <= n implies Comput (P1,s,\$1) = Comput (P2,s,\$1) );
A3: for m being Nat st S1[m] holds
S1[m + 1]
proof
let m be Nat; :: thesis: ( S1[m] implies S1[m + 1] )
assume A4: S1[m] ; :: thesis: S1[m + 1]
A5: Comput (P2,s,(m + 1)) = Following (P2,(Comput (P2,s,m))) by EXTPRO_1:3
.= Exec ((CurInstr (P2,(Comput (P2,s,m)))),(Comput (P2,s,m))) ;
A6: Comput (P1,s,(m + 1)) = Following (P1,(Comput (P1,s,m))) by EXTPRO_1:3
.= Exec ((CurInstr (P1,(Comput (P1,s,m)))),(Comput (P1,s,m))) ;
assume A7: m + 1 <= n ; :: thesis: Comput (P1,s,(m + 1)) = Comput (P2,s,(m + 1))
then m < n by NAT_1:13;
then A8: IC (Comput (P1,s,m)) = IC (Comput (P2,s,m)) by A4;
m < n by ;
then A9: IC (Comput (P2,s,m)) in dom I by A2;
dom P2 = NAT by PARTFUN1:def 2;
then A10: IC (Comput (P2,s,m)) in dom P2 ;
dom P1 = NAT by PARTFUN1:def 2;
then IC (Comput (P1,s,m)) in dom P1 ;
then CurInstr (P1,(Comput (P1,s,m))) = P1 . (IC (Comput (P1,s,m))) by PARTFUN1:def 6
.= I . (IC (Comput (P1,s,m))) by
.= P2 . (IC (Comput (P2,s,m))) by
.= CurInstr (P2,(Comput (P2,s,m))) by ;
hence Comput (P1,s,(m + 1)) = Comput (P2,s,(m + 1)) by A4, A6, A5, A7, NAT_1:13; :: thesis: verum
end;
A11: S1[ 0 ] ;
thus for m being Nat holds S1[m] from NAT_1:sch 2(A11, A3); :: thesis: verum