let N be non empty with_non-empty_elements set ; :: thesis: for n being Element of NAT
for S being non empty IC-Ins-separated AMI-Struct of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Element of NAT st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)
let n be Element of NAT ; :: thesis: for S being non empty IC-Ins-separated AMI-Struct of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Element of NAT st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m)
let S be non empty IC-Ins-separated AMI-Struct of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Element of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Element of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) holds
for m being Element of 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 Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ) implies for m being Element of 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 Element of NAT st
( m < n & not IC (Comput (P2,s,m)) in dom I ) or for m being Element of NAT st m <= n holds
Comput (P1,s,m) = Comput (P2,s,m) )
assume A3: for m being Element of NAT st m < n holds
IC (Comput (P2,s,m)) in dom I ; :: thesis: for m being Element of 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) );
A4: for m being Element of NAT st S₁[m] holds
S₁[m + 1]

proof

let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A5: S₁[m] ; :: thesis: S₁[m + 1]
A6: 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))) ;
A7: 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 A8: m + 1 <= n ; :: thesis: Comput (P1,s,(m + 1)) = Comput (P2,s,(m + 1))
then m < n by NAT_1:13;
then A9: IC (Comput (P1,s,m)) = IC (Comput (P2,s,m)) by A5;
m < n by A8, NAT_1:13;
then A10: IC (Comput (P2,s,m)) in dom I by A3;
dom P2 = NAT by PARTFUN1:def 2;
then A11: 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 A10, A9, A1, GRFUNC_1:2
.= P2 . (IC (Comput (P2,s,m))) by A10, A9, A1, GRFUNC_1:2
.= CurInstr (P2,(Comput (P2,s,m))) by A11, PARTFUN1:def 6 ;
hence Comput (P1,s,(m + 1)) = Comput (P2,s,(m + 1)) by A5, A7, A6, A8, NAT_1:13; :: thesis: verum

end;

Comput (P1,s,0) = s by EXTPRO_1:2;
then A12: S₁[ 0 ] by EXTPRO_1:2;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A12, A4); :: thesis: verum