let s be State of SCMPDS; :: thesis: for P being Instruction-Sequence of SCMPDS
for I being Program of
for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let P be Instruction-Sequence of SCMPDS; :: thesis: for I being Program of
for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let I be Program of ; :: thesis: for k being Nat st I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) holds
IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
let k be Nat; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P & k < LifeSpan ((P +* (stop I)),(Initialize s)) implies IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I )
set ss = Initialize s;
set PP = P +* (stop I);
set m = LifeSpan ((P +* (stop I)),(Initialize s));
set Sp = Stop SCMPDS;
assume that
A1: I is_closed_on s,P and
A2: I is_halting_on s,P and
A3: k < LifeSpan ((P +* (stop I)),(Initialize s)) ; :: thesis: IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I
set Sk = Comput ((P +* (stop I)),(Initialize s),k);
set Ik = IC (Comput ((P +* (stop I)),(Initialize s),k));
A4: IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom (stop I) by A1;
reconsider n = IC (Comput ((P +* (stop I)),(Initialize s),k)) as Nat ;
A5: stop I c= P +* (stop I) by FUNCT_4:25;
A6: P +* (stop I) halts_on Initialize s by A2;

A7: now :: thesis: not n = card I

A8: (P +* (stop I)) /. (IC (Comput ((P +* (stop I)),(Initialize s),k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by PBOOLE:143;
assume A9: n = card I ; :: thesis: contradiction
CurInstr ((P +* (stop I)),(Comput ((P +* (stop I)),(Initialize s),k))) = (P +* (stop I)) . (IC (Comput ((P +* (stop I)),(Initialize s),k))) by A8
.= (stop I) . (0 + n) by A4, A5, GRFUNC_1:2
.= halt SCMPDS by A9, Lm1, Lm2, AFINSQ_1:def 3 ;
hence contradiction by A3, A6, EXTPRO_1:def 15; :: thesis: verum

end;

card (stop I) = (card I) + 1 by COMPOS_1:55;
then n < (card I) + 1 by A4, AFINSQ_1:66;
then n <= card I by INT_1:7;
then n < card I by A7, XXREAL_0:1;
hence IC (Comput ((P +* (stop I)),(Initialize s),k)) in dom I by AFINSQ_1:66; :: thesis: verum