let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being 0 -started State of SCMPDS
for I being parahalting Program of SCMPDS
for J being Program of SCMPDS st stop I c= P holds
for m being Element of NAT st m <= LifeSpan (P,s) holds
NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m))
let s be 0 -started State of SCMPDS; :: thesis: for I being parahalting Program of SCMPDS
for J being Program of SCMPDS st stop I c= P holds
for m being Element of NAT st m <= LifeSpan (P,s) holds
NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m))
let I be parahalting Program of SCMPDS; :: thesis: for J being Program of SCMPDS st stop I c= P holds
for m being Element of NAT st m <= LifeSpan (P,s) holds
NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m))
let J be Program of SCMPDS; :: thesis: ( stop I c= P implies for m being Element of NAT st m <= LifeSpan (P,s) holds
NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m)) )
set SI = stop I;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (P,s) implies NPP (Comput (P,s,$1)) = NPP (Comput ((P +* (I ';' J)),s,$1)) );
A1: Comput ((P +* (I ';' J)),s,0) = s by EXTPRO_1:3;
assume A2: stop I c= P ; :: thesis: for m being Element of NAT st m <= LifeSpan (P,s) holds
NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m))
then A3: P halts_on s by SCMPDS_4:def 10;
A4: for m being Element of 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 A5: dom I c= dom (I ';' J) by XBOOLE_1:7;
let m be Element of NAT ; :: thesis: ( S₁[m] implies S₁[m + 1] )
assume A6: ( m <= LifeSpan (P,s) implies NPP (Comput (P,s,m)) = NPP (Comput ((P +* (I ';' J)),s,m)) ) ; :: thesis: S₁[m + 1]
assume A7: m + 1 <= LifeSpan (P,s) ; :: thesis: NPP (Comput (P,s,(m + 1))) = NPP (Comput ((P +* (I ';' J)),s,(m + 1)))
then A8: IC (Comput (P,s,m)) = IC (Comput ((P +* (I ';' J)),s,m)) by A6, COMPOS_1:230, NAT_1:13;
A10: Comput ((P +* (I ';' J)),s,(m + 1)) = Following ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by EXTPRO_1:4
.= Exec ((CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m)))),(Comput ((P +* (I ';' J)),s,m))) ;
A11: Comput (P,s,(m + 1)) = Following (P,(Comput (P,s,m))) by EXTPRO_1:4
.= Exec ((CurInstr (P,(Comput (P,s,m)))),(Comput (P,s,m))) ;
A12: I ';' J c= P +* (I ';' J) by FUNCT_4:26;
A13: IC (Comput (P,s,m)) in dom (stop I) by A2, SCMPDS_4:def 9;
A15: P /. (IC (Comput (P,s,m))) = P . (IC (Comput (P,s,m))) by PBOOLE:158;
stop I c= P by A2;
then A16: CurInstr (P,(Comput (P,s,m))) = (stop I) . (IC (Comput (P,s,m))) by A13, A15, GRFUNC_1:8;
A17: (P +* (I ';' J)) /. (IC (Comput ((P +* (I ';' J)),s,m))) = (P +* (I ';' J)) . (IC (Comput ((P +* (I ';' J)),s,m))) by PBOOLE:158;
m < LifeSpan (P,s) by A7, NAT_1:13;
then (stop I) . (IC (Comput (P,s,m))) <> halt SCMPDS by A3, A16, EXTPRO_1:def 14;
then A18: IC (Comput (P,s,m)) in dom I by A13, Th3;
then A19: IC (Comput (P,s,m)) in dom I ;
then CurInstr (P,(Comput (P,s,m))) = I . (IC (Comput (P,s,m))) by A16, AFINSQ_1:def 4
.= (I ';' J) . (IC (Comput (P,s,m))) by A19, AFINSQ_1:def 4
.= CurInstr ((P +* (I ';' J)),(Comput ((P +* (I ';' J)),s,m))) by A8, A12, A18, A5, A17, GRFUNC_1:8 ;
hence NPP (Comput (P,s,(m + 1))) = NPP (Comput ((P +* (I ';' J)),s,(m + 1))) by A6, A7, A11, A10, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

Comput (P,s,0) = s by EXTPRO_1:3;
then A20: S₁[ 0 ] by A1;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A20, A4); :: thesis: verum