let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS
for I being parahalting Program of
for J being Program of st stop I c= P holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m)
let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of
for J being Program of st stop I c= P holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m)
let I be parahalting Program of ; :: thesis: for J being Program of st stop I c= P holds
for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m)
let J be Program of ; :: thesis: ( stop I c= P implies for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m) )
set SI = stop I;
defpred S₁[ Nat] means ( $1 <= LifeSpan (P,s) implies Comput (P,s,$1) = Comput ((P +* (I ';' J)),s,$1) );
assume A1: stop I c= P ; :: thesis: for m being Nat st m <= LifeSpan (P,s) holds
Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m)
then A2: P halts_on s by SCMPDS_4:def 7;
A3: for m being Nat st S₁[m] holds
S₁[m + 1]

proof

dom (I ';' J) = (dom I) \/ (dom (Shift (J,(card I)))) by FUNCT_4:def 1;
then A4: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Nat; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A5: ( m <= LifeSpan (P,s) implies Comput (P,s,m) = Comput ((P +* (I ';' J)),s,m) ) ; :: thesis: S₁[m + 1]
assume A6: m + 1 <= LifeSpan (P,s) ; :: thesis: Comput (P,s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1))
A7: Comput ((P +* (I ';' J)),s,(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by EXTPRO_1:3;
A8: Comput (P,s,(m + 1)) = Following (P,(Comput (P,s,m))) by EXTPRO_1:3;
A9: I ';' J c= P +* (I ';' J) by FUNCT_4:25;
A10: IC (Comput (P,s,m)) in dom (stop I) by A1, SCMPDS_4:def 6;
A11: P /. (IC (Comput (P,s,m))) = P . (IC (Comput (P,s,m))) by PBOOLE:143;
A12: CurInstr (P,(Comput (P,s,m))) = (stop I) . (IC (Comput (P,s,m))) by A10, A11, A1, GRFUNC_1:2;
A13: (P +* (I ';' J)) /. (IC (Comput ((P +* (I ';' J)),s,m))) = (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),s,m))) by PBOOLE:143;
m < LifeSpan (P,s) by A6, NAT_1:13;
then (stop I) . (IC (Comput (P,s,m))) <> halt SCMPDS by A2, A12, EXTPRO_1:def 15;
then A14: IC (Comput (P,s,m)) in dom I by A10, COMPOS_1:51;
CurInstr (P,(Comput (P,s,m))) = I . (IC (Comput (P,s,m))) by A12, A14, AFINSQ_1:def 3
.= (I ';' J) . (IC (Comput (P,s,m))) by A14, AFINSQ_1:def 3
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by A6, A9, A14, A4, A13, A5, GRFUNC_1:2, NAT_1:13 ;
hence Comput (P,s,(m + 1)) = Comput ((P +* (I ';' J)),s,(m + 1)) by A5, A6, A8, A7, NAT_1:13; :: thesis: verum

end;

A15: S₁[ 0 ] ;
thus for m being Nat holds S₁[m] from NAT_1:sch 2(A15, A3); :: thesis: verum