let P be the Instructions of SCMPDS -valued ManySortedSet of NAT ; :: thesis: for s being State of SCMPDS
for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let s be State of SCMPDS; :: thesis: for I being halt-free Program of SCMPDS
for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let I be halt-free Program of SCMPDS; :: thesis: for J being Program of SCMPDS st I is_closed_on s,P & I is_halting_on s,P holds
IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
let J be Program of SCMPDS; :: thesis: ( I is_closed_on s,P & I is_halting_on s,P implies IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1 )
set m = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1;
set G = Goto ((card J) + 1);
set s2 = Initialize s;
set P2 = P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J));
assume A1: ( I is_closed_on s,P & I is_halting_on s,P ) ; :: thesis: IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1
then ( P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J)) halts_on Initialize s & LifeSpan ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s)) = (LifeSpan ((P +* (stop I)),(Initialize s))) + 1 ) by Lm3;
then IC (Result ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s))) = IC (Comput ((P +* (stop ((I ';' (Goto ((card J) + 1))) ';' J))),(Initialize s),((LifeSpan ((P +* (stop I)),(Initialize s))) + 1))) by EXTPRO_1:23
.= ((card I) + (card J)) + 1 by A1, Lm3 ;
hence IC (IExec (((I ';' (Goto ((card J) + 1))) ';' J),P,s)) = ((card I) + (card J)) + 1 by SCMPDS_5:22; :: thesis: verum