let a be Int_position; :: thesis: for P being Instruction-Sequence of SCMPDS

for s being 0 -started State of SCMPDS

for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS

for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let I be halt-free parahalting Program of ; :: thesis: for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let J be parahalting shiftable Program of ; :: thesis: (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

A1: not a in dom (Start-At (((IC (IExec (J,P,(Initialize (IExec (I,P,s)))))) + (card I)),SCMPDS)) by SCMPDS_4:18;

IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(Initialize (IExec (I,P,s))))),(card I)) by Th22;

hence (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a by A1, FUNCT_4:11; :: thesis: verum

for s being 0 -started State of SCMPDS

for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let P be Instruction-Sequence of SCMPDS; :: thesis: for s being 0 -started State of SCMPDS

for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let s be 0 -started State of SCMPDS; :: thesis: for I being halt-free parahalting Program of

for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let I be halt-free parahalting Program of ; :: thesis: for J being parahalting shiftable Program of holds (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

let J be parahalting shiftable Program of ; :: thesis: (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a

A1: not a in dom (Start-At (((IC (IExec (J,P,(Initialize (IExec (I,P,s)))))) + (card I)),SCMPDS)) by SCMPDS_4:18;

IExec ((I ';' J),P,s) = IncIC ((IExec (J,P,(Initialize (IExec (I,P,s))))),(card I)) by Th22;

hence (IExec ((I ';' J),P,s)) . a = (IExec (J,P,(Initialize (IExec (I,P,s))))) . a by A1, FUNCT_4:11; :: thesis: verum