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= s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
let I be parahalting Program of SCMPDS; :: thesis: for J being Program of SCMPDS st stop I c= s holds
for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
let J be Program of SCMPDS; :: thesis: ( stop I c= s implies for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT )
set SI = stop I;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan ((ProgramPart s),s) implies Comput ((ProgramPart s),s,$1), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),$1) equal_outside NAT );
A1: Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),0) = s +* (I ';' J) by EXTPRO_1:3;
assume A2: stop I c= s ; :: thesis: for m being Element of NAT st m <= LifeSpan ((ProgramPart s),s) holds
Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT
then A3: ProgramPart s 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 ((ProgramPart s),s) implies Comput ((ProgramPart s),s,m), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) equal_outside NAT ) ; :: thesis: S₁[m + 1]
assume A7: m + 1 <= LifeSpan ((ProgramPart s),s) ; :: thesis: Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) equal_outside NAT
then A8: IC (Comput ((ProgramPart s),s,m)) = IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) by A6, COMPOS_1:24, NAT_1:13;
T1: ProgramPart (s +* (I ';' J)) = ProgramPart (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) by AMI_1:123;
A9: Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) = Following ((ProgramPart (s +* (I ';' J))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart (s +* (I ';' J))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) ;
A10: Comput ((ProgramPart s),s,(m + 1)) = Following ((ProgramPart s),(Comput ((ProgramPart s),s,m))) by EXTPRO_1:4
.= Exec ((CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m)))),(Comput ((ProgramPart s),s,m))) ;
A11: I ';' J c= Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m) by AMI_1:81, FUNCT_4:26;
A12: IC (Comput ((ProgramPart s),s,m)) in dom (stop I) by A2, SCMPDS_4:def 9;
TX: ProgramPart s = ProgramPart (Comput ((ProgramPart s),s,m)) by AMI_1:123;
Y: (ProgramPart s) /. (IC (Comput ((ProgramPart s),s,m))) = (Comput ((ProgramPart s),s,m)) . (IC (Comput ((ProgramPart s),s,m))) by TX, COMPOS_1:38;
stop I c= Comput ((ProgramPart s),s,m) by A2, AMI_1:81;
then A13: CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = (stop I) . (IC (Comput ((ProgramPart s),s,m))) by A12, Y, GRFUNC_1:8;
Y: (ProgramPart (s +* (I ';' J))) /. (IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) = (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m)) . (IC (Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by T1, COMPOS_1:38;
m < LifeSpan ((ProgramPart s),s) by A7, NAT_1:13;
then (stop I) . (IC (Comput ((ProgramPart s),s,m))) <> halt SCMPDS by A3, A13, EXTPRO_1:def 14;
then A14: IC (Comput ((ProgramPart s),s,m)) in dom I by A12, Th3;
then XX: IC (Comput ((ProgramPart s),s,m)) in dom I ;
then CurInstr ((ProgramPart s),(Comput ((ProgramPart s),s,m))) = I . (IC (Comput ((ProgramPart s),s,m))) by A13, AFINSQ_1:def 4
.= (I ';' J) . (IC (Comput ((ProgramPart s),s,m))) by XX, AFINSQ_1:def 4
.= CurInstr ((ProgramPart (s +* (I ';' J))),(Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),m))) by A8, A11, A14, A5, Y, GRFUNC_1:8 ;
hence Comput ((ProgramPart s),s,(m + 1)), Comput ((ProgramPart (s +* (I ';' J))),(s +* (I ';' J)),(m + 1)) equal_outside NAT by A6, A7, A10, A9, NAT_1:13, SCMPDS_4:15; :: thesis: verum

end;

Comput ((ProgramPart s),s,0) = s by EXTPRO_1:3;
then A15: S₁[ 0 ] by A1, FUNCT_7:132;
thus for m being Element of NAT holds S₁[m] from NAT_1:sch 1(A15, A4); :: thesis: verum