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 (Comput ((ProgramPart s2),s2,m)) in dom I ) holds
for m being Element of NAT st m <= n holds
Comput ((ProgramPart s1),s1,m), Comput ((ProgramPart s2),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 (Comput ((ProgramPart s2),s2,m)) in dom I ) holds
for m being Element of NAT st m <= n holds
Comput ((ProgramPart s1),s1,m), Comput ((ProgramPart s2),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 (Comput ((ProgramPart s2),s2,m)) in dom I ) implies for m being Element of NAT st m <= n holds
Comput ((ProgramPart s1),s1,m), Comput ((ProgramPart s2),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 (Comput ((ProgramPart s2),s2,m)) in dom I ; :: thesis: for m being Element of NAT st m <= n holds
Comput ((ProgramPart s1),s1,m), Comput ((ProgramPart s2),s2,m) equal_outside NAT
defpred S₁[ Nat] means ( $1 <= n implies Comput ((ProgramPart s1),s1,$1), Comput ((ProgramPart s2),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 Comput ((ProgramPart s1),s1,m), Comput ((ProgramPart s2),s2,m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
A7: Comput ((ProgramPart s2),s2,(m + 1)) = Following ((ProgramPart s2),(Comput ((ProgramPart s2),s2,m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,m))),(Comput ((ProgramPart s2),s2,m)))),(Comput ((ProgramPart s2),s2,m))) by AMI_1:123 ;
A8: Comput ((ProgramPart s1),s1,(m + 1)) = Following ((ProgramPart s1),(Comput ((ProgramPart s1),s1,m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,m))),(Comput ((ProgramPart s1),s1,m)))),(Comput ((ProgramPart s1),s1,m))) by AMI_1:123 ;
assume A9: m + 1 <= n ; :: thesis: Comput ((ProgramPart s1),s1,(m + 1)), Comput ((ProgramPart s2),s2,(m + 1)) equal_outside NAT
then A10: IC (Comput ((ProgramPart s1),s1,m)) = IC (Comput ((ProgramPart s2),s2,m)) by A6, COMPOS_1:24, NAT_1:13;
m < n by A9, NAT_1:13;
then A11: IC (Comput ((ProgramPart s2),s2,m)) in dom I by A4;
Y: (ProgramPart (Comput ((ProgramPart s1),s1,m))) /. (IC (Comput ((ProgramPart s1),s1,m))) = (Comput ((ProgramPart s1),s1,m)) . (IC (Comput ((ProgramPart s1),s1,m))) by COMPOS_1:38;
Z: (ProgramPart (Comput ((ProgramPart s2),s2,m))) /. (IC (Comput ((ProgramPart s2),s2,m))) = (Comput ((ProgramPart s2),s2,m)) . (IC (Comput ((ProgramPart s2),s2,m))) by COMPOS_1:38;
CurInstr ((ProgramPart (Comput ((ProgramPart s1),s1,m))),(Comput ((ProgramPart s1),s1,m))) = s1 . (IC (Comput ((ProgramPart s1),s1,m))) by Y, AMI_1:54
.= I . (IC (Comput ((ProgramPart s1),s1,m))) by A2, A11, A10, GRFUNC_1:8
.= s2 . (IC (Comput ((ProgramPart s2),s2,m))) by A3, A11, A10, GRFUNC_1:8
.= CurInstr ((ProgramPart (Comput ((ProgramPart s2),s2,m))),(Comput ((ProgramPart s2),s2,m))) by Z, AMI_1:54 ;
hence Comput ((ProgramPart s1),s1,(m + 1)), Comput ((ProgramPart s2),s2,(m + 1)) equal_outside NAT by A6, A8, A7, A9, Th15, NAT_1:13; :: thesis: verum

end;

Comput ((ProgramPart s1),s1,0) = s1 by EXTPRO_1:3;
then A12: S₁[ 0 ] by A1, EXTPRO_1:3;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A12, A5); :: thesis: verum