let n be Element of NAT ; :: thesis: for I being Program of SCMPDS
for s1, s2 being State of SCMPDS st s1,s2 equal_outside NAT & I c= s1 & I c= s2 & ( for m being Element of NAT st m < n holds
IC (Computation s2,m) in dom I ) holds
for m being Element of NAT st m <= n holds
Computation s1,m, Computation s2,m equal_outside NAT
let I be Program of SCMPDS ; :: thesis: for s1, s2 being State of SCMPDS st s1,s2 equal_outside NAT & I c= s1 & I c= s2 & ( for m being Element of NAT st m < n holds
IC (Computation s2,m) in dom I ) holds
for m being Element of NAT st m <= n holds
Computation s1,m, Computation s2,m equal_outside NAT
let s1, s2 be State of SCMPDS ; :: thesis: ( s1,s2 equal_outside NAT & I c= s1 & I c= s2 & ( for m being Element of NAT st m < n holds
IC (Computation s2,m) in dom I ) implies for m being Element of NAT st m <= n holds
Computation s1,m, Computation s2,m equal_outside NAT )
assume that
A1: s1,s2 equal_outside NAT and
A2: I c= s1 and
A3: I c= s2 and
A4: for m being Element of NAT st m < n holds
IC (Computation s2,m) in dom I ; :: thesis: for m being Element of NAT st m <= n holds
Computation s1,m, Computation s2,m equal_outside NAT
defpred S₁[ Nat] means ( $1 <= n implies Computation s1,$1, Computation s2,$1 equal_outside NAT );
A5: 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 A6: ( m <= n implies Computation s1,m, Computation s2,m equal_outside NAT ) ; :: thesis: S₁[m + 1]
A7: Computation s2,(m + 1) = Following (Computation s2,m) by AMI_1:14
.= Exec (CurInstr (Computation s2,m)),(Computation s2,m) ;
A8: Computation s1,(m + 1) = Following (Computation s1,m) by AMI_1:14
.= Exec (CurInstr (Computation s1,m)),(Computation s1,m) ;
assume A9: m + 1 <= n ; :: thesis: Computation s1,(m + 1), Computation s2,(m + 1) equal_outside NAT
then A10: IC (Computation s1,m) = IC (Computation s2,m) by A6, AMI_1:121, NAT_1:13;
m < n by A9, NAT_1:13;
then A11: IC (Computation s2,m) in dom I by A4;
CurInstr (Computation s1,m) = s1 . (IC (Computation s1,m)) by AMI_1:54
.= I . (IC (Computation s1,m)) by A2, A11, A10, GRFUNC_1:8
.= s2 . (IC (Computation s2,m)) by A3, A11, A10, GRFUNC_1:8
.= CurInstr (Computation s2,m) by AMI_1:54 ;
hence Computation s1,(m + 1), Computation s2,(m + 1) equal_outside NAT by A6, A8, A7, A9, Th15, NAT_1:13; :: thesis: verum

end;

Computation s1,0 = s1 by AMI_1:13;
then A12: S₁[ 0 ] by A1, AMI_1:13;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A12, A5); :: thesis: verum