let s be State of SCMPDS ; :: thesis: for I, J being Program of SCMPDS
for k being Element of NAT st I c= J & I is_closed_on s & I is_halting_on s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
let I, J be Program of SCMPDS ; :: thesis: for k being Element of NAT st I c= J & I is_closed_on s & I is_halting_on s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) holds
Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
let k be Element of NAT ; :: thesis: ( I c= J & I is_closed_on s & I is_halting_on s & k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT )
set m = LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I));
assume that
A1: I c= J and
A2: I is_closed_on s and
A3: I is_halting_on s and
A4: k <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT
set s1 = (Initialize s) +* J;
set s2 = (Initialize s) +* (stop I);
I1: s +* (Initialize J) = (Initialize s) +* J by SCMPDS_4:5;
I2: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by SCMPDS_4:5;
defpred S₁[ Element of NAT ] means ( $1 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) implies Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),$1, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),$1 equal_outside NAT );
A5: dom I c= dom J by A1, GRFUNC_1:8;

A6: now

let k be Element of NAT ; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A7: S₁[k] ; :: thesis: S₁[k + 1]

now

T: ProgramPart ((Initialize s) +* (stop I)) = ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by AMI_1:123;
A8: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(k + 1) = Following (ProgramPart ((Initialize s) +* (stop I))),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by T ;
T: ProgramPart ((Initialize s) +* J) = ProgramPart (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) by AMI_1:123;
A9: Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),(k + 1) = Following (ProgramPart ((Initialize s) +* J)),(Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) by AMI_1:14
.= Exec (CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)),(Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)),(Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) by T ;
A10: k < k + 1 by XREAL_1:31;
assume A11: k + 1 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),(k + 1), Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(k + 1) equal_outside NAT
then k < LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) by A10, XXREAL_0:2;
then A12: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom I by A2, A3, SCMPDS_6:40;
then A13: IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) in dom (stop I) by FUNCT_4:13;
Y: (ProgramPart (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) /. (IC (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) = (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) . (IC (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)) by COMPOS_1:38;
Z: (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) /. (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) = (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by COMPOS_1:38;
CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k)),(Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) = (Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A7, A11, A10, Y, COMPOS_1:24, XXREAL_0:2
.= (s +* (Initialize J)) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by I1, AMI_1:54
.= J . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A5, A12, Th10, I1
.= I . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A1, A12, GRFUNC_1:8
.= (stop I) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A12, SCMPDS_4:37
.= (s +* (Initialize (stop I))) . (IC (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)) by A13, Th10, I2
.= CurInstr (ProgramPart (Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k)),(Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k) by Z, I2, AMI_1:54 ;
hence Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),(k + 1), Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),(k + 1) equal_outside NAT by A7, A11, A10, A9, A8, SCMPDS_4:15, XXREAL_0:2; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A14: S₁[ 0 ]

proof

assume 0 <= LifeSpan (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)) ; :: thesis: Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),0 , Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 equal_outside NAT
A15: Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 = (Initialize s) +* (stop I) by AMI_1:13;
Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),0 = (Initialize s) +* J by AMI_1:13;
hence Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),0 , Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),0 equal_outside NAT by A15, FUNCT_7:134; :: thesis: verum

end;

for k being Element of NAT holds S₁[k] from NAT_1:sch 1(A14, A6);
hence Comput (ProgramPart ((Initialize s) +* J)),((Initialize s) +* J),k, Comput (ProgramPart ((Initialize s) +* (stop I))),((Initialize s) +* (stop I)),k equal_outside NAT by A4; :: thesis: verum