let s be State of SCMPDS ; :: thesis: for I being Program of SCMPDS
for i being Element of NAT st Initialize (stop I) c= s & I is_closed_on s & I is_halting_on s & i < LifeSpan (ProgramPart s),s holds
IC (Comput (ProgramPart s),s,i) in dom I
let I be Program of SCMPDS ; :: thesis: for i being Element of NAT st Initialize (stop I) c= s & I is_closed_on s & I is_halting_on s & i < LifeSpan (ProgramPart s),s holds
IC (Comput (ProgramPart s),s,i) in dom I
let i be Element of NAT ; :: thesis: ( Initialize (stop I) c= s & I is_closed_on s & I is_halting_on s & i < LifeSpan (ProgramPart s),s implies IC (Comput (ProgramPart s),s,i) in dom I )
set sI = stop I;
set Ci = Comput (ProgramPart s),s,i;
set Lc = IC (Comput (ProgramPart s),s,i);
assume that
A1: Initialize (stop I) c= s and
A2: I is_closed_on s and
A3: I is_halting_on s and
A4: i < LifeSpan (ProgramPart s),s ; :: thesis: IC (Comput (ProgramPart s),s,i) in dom I
I1: s +* (Initialize (stop I)) = (Initialize s) +* (stop I) by SCMPDS_4:5;
A5: s = (Initialize s) +* (stop I) by A1, I1, FUNCT_4:79;
then A6: IC (Comput (ProgramPart s),s,i) in dom (stop I) by A2, SCMPDS_6:def 2;
stop I c= Initialize (stop I) by SCMPDS_4:9;
then A7: stop I c= s by A1, XBOOLE_1:1;
A8: ProgramPart s halts_on s by A3, A5, SCMPDS_6:def 3;

now

assume A9: (stop I) . (IC (Comput (ProgramPart s),s,i)) = halt SCMPDS ; :: thesis: contradiction
TX: ProgramPart (Comput (ProgramPart s),s,i) = ProgramPart s by AMI_1:123;
Y: (ProgramPart (Comput (ProgramPart s),s,i)) /. (IC (Comput (ProgramPart s),s,i)) = (Comput (ProgramPart s),s,i) . (IC (Comput (ProgramPart s),s,i)) by COMPOS_1:38;
CurInstr (ProgramPart (Comput (ProgramPart s),s,i)),(Comput (ProgramPart s),s,i) = s . (IC (Comput (ProgramPart s),s,i)) by Y, AMI_1:54
.= halt SCMPDS by A6, A7, A9, GRFUNC_1:8 ;
hence contradiction by A4, A8, TX, AMI_1:def 46; :: thesis: verum

end;

hence IC (Comput (ProgramPart s),s,i) in dom I by A6, SCMPDS_5:3; :: thesis: verum